Since everyone else seems to have removed their comments I will have a go. There are really two questions here. Why did Elizabeth not get married and provide an heir of her own body and when that became impracticable did she just not name someone with a convincing claim to the throne as heir.

There are a series of answers to the first question. Any marriage that she might make came with serious political issues. As a wife she would have to give up some of her power to her husband, as Mary had done to Philip II. Elizabeth was extremely attached to her own prerogatives was not at all enthused by the idea of giving up her power. Further, any foreign marriage would come with alliances attached, which the English might well not accept. An English match also posed problems because of status conflicts and the fact that the preferred candidate, Dudley, was married (his wife died under mysterious circumstances and suspicions that she was pushed down the stairs were rife). Further, Elizabeth gained political advantage from seeming to entertain marriage while always refusing to get pinned down. So she entertained the proposals of Eric of Sweden, Charles of Austria and Henry and Francis of Anjou. Always the possibility of alliance was dangled and this helped to keep relations with these countries sweet while marriage negotiations were ongoing. There was also the fact that any marriage that she might make would come with political consequences. The developing Puritan faction would not accept a Catholic match, with all the consequences for English protestantism which that might entail, and any misstep might cause insurrection. Finally, if she did produce an heir there was no reason that she might not just be killed and her infant child placed on the throne with a protectorate run by her husband. In many ways she was quite effectively boxed in and decided that doing nothing regarding marriage was the best course.

As for naming a successor after she passed menopause, again this posed massive difficulties for her. Probably the person with the best claim was Mary Queen of Scots. The issue was that Mary was a) Catholic and b) in exile in England and stirring up trouble amongst the northern earls (the Northern Rising of 1569 was partially designed to get Mary on the throne). Any move to make Mary heir would inflame the Protestants against her and give the Catholics cause to just have her assassinated in the knowledge that Mary would end up on the throne. After the execution of Mary it seemed that the best course was simply to wait and hope than ambiguity would keep all the factions around her quiet. The Puritans could expect that James of Scotland would become king while the Catholics could hope for someone else. This actually quite well for her and the last years of her reign, despite economic recession and massive debt were relatively quiet. It also helped that Cecil was manoeuvring behind Elizabeth's back to ensure that James would come to the throne peacefully.

Asking the question of why Elizabeth I did not name a successor raises a series of wider questions. Who would have been a suitable successor for the English throne in 1603? Or more specifically, what qualities were required for the ideal monarch of England in 1603? Male or female? Catholic or Protestant? Politically experienced or otherwise? English born, or alien? The succession story of 1603 also demonstrates how key the Privy Council, a body that had actually only been formally established as a department of government after the fall of Thomas Cromwell under Henry VIII in 1540, had become to English politics, which is another part of the wider story that underpins this question. My answer, while addressing the original question, must also touch on points such as these to give a full explanation of the Tudor-Stuart succession that occurred in 1603. Once we have accepted the fact that there is no simple answer to who Elizabeth should have named as her successor, we can go into much more detail around the particulars of the 1603 succession.

The relative ease with which James VI of Scotland became James I of England in March 1603 is more remarkable than it is self-evident, and has often led to a lack of acknowledgement of the very real and ongoing succession debate that characterised Elizabeth I's last years as queen. There was an atmosphere of great uncertainty and fear surrounding the death of the queen. The succession question was by no means certain, even at this late stage, and indeed it is abundantly clear that James himself was riddled with doubts over the issue.

The debate over who would succeed was not exclusively housed in England, but concerned several foreign powers on the continent, namely Spain and France, and of course, the Pope in Rome, all of whom were concerned as much with religious as with political matters at a time when political uncertainty in England represented a possibility of Catholic conversion. Elizabeth's refusal, first to provide a successor through the traditional methods of marriage and birth, and then later to name one when prospects of marriage had passed and age had rendered her fertility doubtful, meant that the contemporary political elite were forced to engage with the notion of what gave any candidate the right to become king of England.

Going back through English history to look for precedent, it was by no means certain that hereditary right was the sole, or even necessarily the most important, quality required for assuming the throne. Accession through conquest was at least as legitimate in recent English history: arguably, even the throne of Mary I was one obtained in 1553 by a sort of conquest, though her genealogical claim also placed her in good stead to maintain her position, and indeed in a position to launch her assault on the throne in the first instance.

Through the first half of Elizabeth's reign, it made sense politically for her not to name an heir, as doing so would run the risk of providing a figurehead for those disenchanted with Elizabethan rule to rally behind, which could spark rebellion. Yet Elizabeth had the advantage of outliving the two biggest threats to her throne, Catherine Grey and Mary Stuart, and in the last years of her reign it would perhaps have been prudent to name an heir and thus save any political complications arising upon her death. Her failure to do so, though, meant that Robert Cecil took into his own hands the task of securing the throne for James VI. Yet James of Scotland was not the only candidate considered worthy of the throne.

So what qualities rendered the discussed candidates suitable for consideration for the English throne in 1603? Wilson took a merely genealogical approach to the situation, outlining the various blood claims of his candidates in order of their traditional placing in line to the throne. Proving the blood claim of any candidate was certainly seen as being of imperative importance to contemporaries, with the Jesuit Robert Parsons going to great lengths to prove, or to fabricate proof, that the blood line of the Infanta Isabella of Spain gave her a legitimate claim to the English throne. In 1603, though, the fact is that none of the candidates outlined were entirely suitable. Whilst it is clear that James VI of Scotland held by far the superior blood claim to the throne, if one takes into account the stipulations set out in the will of Henry VIII, James was in fact legally barred from taking the English throne, as was Wilson's second candidate, his cousin, Lady Arabella Stuart.

It should be acknowledged that Edward VI, too, attempted to alter the traditional course of succession in his will, but the failure in 1553 to place Lady Jane Grey on the throne as he stipulated renders his attempt obsolete. The terms of Henry's will, however, still largely stood in place in 1603, and were thus of key importance in the succession struggle. James's riposte to the threat the will posed to his claim stemmed from the theory of the divine right of kings, countering that he claimed the English throne through the power of hereditary right and primogeniture, laws which the will of Henry VIII simply had no power to overturn.

The blood claim of Beauchamp was actually stronger than the blood claim of James VI of Scotland if we take the will of Henry VIII as valid: it was, after all, Henry's will that had dictated the course of succession since his death. Whether it was legally the monarch's prerogative to choose her successor was subject to much debate, but many of Elizabeth’s contemporaries believed that to be the case, urging Elizabeth to speak her opinion on the matter of succession. This method of choosing a successor certainly had precedent throughout the sixteenth century, with Henry VIII and Edward VI both trying to dictate and even pervert the course of succession upon their death through succession acts and stipulations in their wills.

It was reported that as Elizabeth lay on her deathbed, her counsellors ran through a list of potential candidates for the succession and asked her to lift a finger if she supported one. The fact that she did not react to any of the names except for that of Lord Beauchamp, when she reportedly burst into life and fumed that 'I will have no rascal's son in my seat, but one worthy to be a king' probably held much weight amongst those in the Privy Council to whom the decision would at least initially effectively fall in the wake of her death.

The succession story of 1603 thus gives a fascinating insight into the lasting changes that the Tudor monarchs had wrought on English politics. A more meritocratic political system had developed, whereby men such as Robert Cecil were able to vanquish earls in political battle. It demonstrates a move away from feudal systems of government, to one in which the highest ranking members of government took precedence over nobles regardless of their rank. This was no democratic decision, but never before had the choice of monarch relied so heavily upon the personal preference of his subjects. James, with his years of political experience, his even-handed and consistent diplomacy, his awareness of the need for religious compromise, or at the very least ambiguity in rhetoric at a time when religious faction starkly divided English society, had demonstrated himself to be head and shoulders above the rest of his competitors, and it was from this position that he took the crown of England in 1603.

Furthermore, in a mere seventy years, the religious Reformation that had seemed so contentious when first set in motion by Henry VIII had cemented itself to the extent that England had become a state where a Protestant monarch was eminently more desirable than a Catholic one to most elements of society. Here, again, James stood out as the most qualified candidate for the English throne. The turmoil of Mary I’s attempt to reassert Catholicism as the dominant religion of the realm was still relatively recent in the memory of many, rendering the idea of inserting the Spanish Infanta on the English throne as a Catholic monarch as dangerously divisive. Those Protestants concerned with succession in England recognised that it was important to place united support behind one candidate, and as a result Lord Beauchamp became largely overlooked, as James Stuart emerged as the most qualified, and indeed, the most determined Protestant candidate.

This new English Protestantism had vastly wider reaching implications than might initially be assumed. Anne McLaren has demonstrated how, for patriarchal Protestant figures in Elizabethan political life, a female monarch alone was not enough to fill the conceptual role of ‘king’ of England. While Elizabeth, through what was most probably the result of some shrewd manoeuvring of her own, had retained against all precedent monarchical autonomy by refusing to marry and to birth the heir that would give her husband the right of rule, she had also forced those Protestant politicians deeply ingrained in patriarchal models of authority to create new and innovative methods of ensuring that a bodily person other than the titular monarch whom they perceived as ‘king-like’ held authority. McLaren suggests these figures were firstly Robert Dudley, the Earl of Leicester, and then his political heir, the Earl of Essex, though perhaps William Cecil, Lord Burghley, is a more pertinent and enduring example of this sort of power distribution.

Under Elizabeth, then, perhaps the role of ‘king’ had transformed irrevocably. James could proclaim to Parliament all he wanted that ‘precedents in the times of minors, of tyrants, or women or simple kings [are] not to be credited.’ This could not, and indeed did not, invalidate the fact that the monarchical republic that had existed under Elizabeth had paved the way for new forms of power distribution and new modes of governance in England that eventually found full expression in revolution in 1640.

In the documentation of Elizabeth's conversations with William Maitland, the Scots ambassador during the reign of Mary Stuart, she has this to say on the subject of naming Mary (by which we are referring to the House of Stuart) as her heir:

"Yea, (said she) if I meant to do anything to hurt her right, they have occasion to desire me to reform it, but this desire is without an example - to require me in my own life to set my winding-sheet before my eye! The like was never required of no prince." (Source: Elizabeth I: Collected Works)

This is typical of the argument Elizabeth used to defer questions about the succession. What she means by this has two layers: first, that by naming the House of Stuart as her heir, she would be coming too close to clearly stating her intent not to marry and have children of her own. Second, since Mary was Catholic, she would become a locus for plots to depose the Protestant Elizabeth, putting her throne and her life in danger.

There were several points of contention around Elizabeth's reign that are relevant to consider here. One is that, as a woman, her ability to rule was questioned even by those who supported her. (Ex: William Cecil, Lord Burleigh prayed for God to grant England a male succession for its posterity, and he was one of Elizabeth's closest advisors.) Another issue was based around questions regarding Elizabeth's legitimacy. Not everyone accepted her father's divorce from Queen Catherine of Aragon and subsequent marriage to Anne Boleyn as valid. Furthermore, Henry VIII had Elizabeth declared illegitimate by an act of Parliament at one point, and even though he later went back and included her in the line of succession, neither he nor Elizabeth ever formally addressed her status of legitimacy in a legal sense.

My point in all this is that Elizabeth understood that her place on the throne required her to act prudently and not commit herself too much on any account, especially in regards to questions of succession. For this reason, she became a mistress of prevarication and a giver of "answers answerless."

For further reading: Levine's "The Early Elizabethan Succession Question, 1558-1568" and Archer's "Religion, Politics, and Society in 16th Century England."