I've just seen this as someone linked my other answer to it, I think I can answer this question in more detail than some other posters have so I will give this one ago and might copy small parts from my other answer if that's OK!

So, it's very true that one reason Elizabeth did not want to name an heir was to avoid plots against her, and it's also very true that Elizabeth did not want a Catholic (Mary, Queen of Scots) on her throne. Another poster has given some potential ideas in terms of blood claims to the throne, but not all of this is entirely accurate. While it may be true that these people did have a blood claim to the throne, it is absolutely not the case that any of them (bar one or two) were considered likely or suitable heirs to the throne by the Elizabethan political elite. I am also interested by the assumption in your question that 'it would have saved a lot of bother' - a lot of bother for who? It would have caused many people huge amounts of bother to have a Catholic monarch of a foreign country named as heir to the English Protestant throne. It is also true to say that not naming an heir probably saved Elizabeth herself much more bother than it did perhaps ultimately cause for key figures both in the Elizabethan court, as well as those abroad with interests in the English throne.

Let me just say a little more on how the 'Catholic problem' was perceived at the time of succession as it will hopefully provide insight into how important this was. Issues of religion had been a key concern of successive Tudor governments after the upheaval of the Reformation, and the break with Rome under Henry VIII. In Robert Parson’s ‘Conference’, he discusses in detail what he perceives to be the relationship between the monarchy and religion. His suggestion is that the first duty of the monarch is to protect the spirituality of the nation, and that it is therefore of great importance that the monarch and the people are of one religion. 

It was only in very recent European history, after all, that kingdoms had been divided by schism. The last fragment of papal authority in England had only been removed as recently as 1536, when the See of Rome Act was issued. The delicate nature of religious settlement in 1603 was therefore crucial to the debate surrounding succession. Looking back at the respective reigns of Henry VIII, Edward VI, Mary I and Elizabeth I, it had become almost expected that the succession of any new monarch would bring religious upheaval, change, and a new settlement.

Elizabeth died leaving a confused and sometimes contradictory religious policy behind her, and with it the unnerving realisation among her largely Protestant councillors that neither toleration nor persecution seemed to be effective in taming the ever-present 'Catholic problem' that penetrated the realm. The complications of a religiously divided Europe served to further nurture this as an important factor in the English succession debate, as Catholic nations such as France and Spain, as well as the Pope in Rome, had an inherent interest in ensuring that the English monarch, if not Catholic, would at least show tolerance towards his new Catholic subjects. Thus, the period of succession was crucial to any Catholic conspirators both at home and overseas, who had great interest in seeing a Catholic claimant take the throne.

Leanda de Lisle has asserted that 'the English Jesuits, led by their Principal, the Somerset-born Robert Parsons, were the most determined and dangerous opponents of James's succession.' In England, though, it seemed that Catholic factions were practically in a state of war amongst themselves. This divide lay between conciliatory and separatist forces, specifically the secular priests and the Jesuits: whilst the secular priests believed that the execution of Mary, Queen of Scots, had symbolised the end of any hope of a Catholic monarch succeeding Elizabeth and restoring their faith, and instead hoped that by proving their loyalty to the new monarch they would be able to cement their position as an accepted minority, some Jesuits instead continued to push relentlessly for a Catholic monarch to succeed to the throne. 

The Catholic belief in the supreme authority of the Pope over the King of England held firm despite the law of praemunire. For example, the Jesuit Henry Garnet showed a group of former soldiers a secret papal brief which ordered all Catholics to oppose the crowning of a heretic. The increasing persecution of Catholics towards the end of Elizabeth's reign demonstrates the very real fear of a Catholic attempt on the throne that many in government, and indeed, Elizabeth herself, were experiencing. In November 1602, for example, the Privy Council published its 'Proclamation against Jesuits and others', which ordered all Jesuits to leave England within thirty days. So perhaps this illuminates why it would never have made sense for Elizabeth to name a Catholic successor to the throne.

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 (son of Mary QOS). Yet James of Scotland was not the only candidate considered worthy of the throne. 

Going back through English history, 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. 

In 1603, Parliament declared that James took his throne 'by inherent birthright and lawfull and undoubted succession,... beinge lineallie, justly and lawfullie next and sole Heire of the Blood royall of this Realme.' Yet the very fact that it was necessary for the government body of Parliament to declare this suggests another facet to the derivation of the right to be king. Did the right to kingship now stem from parliamentary consent? This being the case, James's claim was surely incontestable, with the support of the English political nation largely planted behind him, paired with the blood claim he himself perceived as indefeasible.

'The State of England Anno Dom. 1600', a work dealing with the succession written by Thomas Wilson, outlined the '12 Competitors that gape for the death of that good old Princess the now Queen', offering to 'sett downe the Pedigrees of all these Competitors and leave the Rest to the Reader to Judge'. Wilson was a figure on the periphery of the Elizabethan court, and his biographer A. F. Pollard sees this work as a ‘pitch for gainful employment.’ As such, he accords closely with Robert Cecil’s views on the succession, and argues at length the merit of James Stuart’s claim to the English throne. He concedes, however, that 'thus you see this Crowne is not like to fall to the ground for want of heads that claime to weare it, but upon whose head it will fall is by many doubted.' This document is probably the best contemporary source we have in terms of summarising the political debate around who should succeed to the English throne after Elizabeth, and gives us a real insight into who was deemed a likely heir at the time. It also gives us a view of the desire of the political elite to have a Protestant or 'Protestant enough' heir to the throne, which in my view is the single most important reason why Mary QOS would never have been named heir.

We don't really know if Elizabeth mentally had a plan for her heir - during her lifetime, she tended not to commit to any possibilities.

Identifying an heir on the Tudor line wasn't the problem, because there were actually a bunch of people who were eligible that way. In addition to Mary Stuart, descended through Margaret Tudor from her first marriage, there were a number of claimants descended from Mary Tudor through her marriage to Charles Brandon: she had two daughters who survived into adulthood, Frances and Eleanor. Frances had three daughters of her own (Jane, Katherine, and Mary Grey), while Eleanor had one child who survived to adulthood (Lady Margaret Clifford), who herself had children - all of these people were actually considered to be in the line of succession after Elizabeth, even if she never held a ceremony and said, "Lady Margaret Clifford is my heir presumptive because the Greys have died out." In addition, there were descendants of Margaret Tudor from her second marriage - including Mary Stuart's husband, Lord Darnley! But a few factors made the whole process extraordinarily complicated.

For one thing, yes, as you mentioned, naming an heir would leave her vulnerable to assassination. This is true of any monarch, but in reality children tend to have emotional ties to their parents, even when raised by nurses and governesses, and so do not usually kill them in order to ascend the throne faster; if a faction that would prefer to see the child on the throne were to exist, they would have to weigh that desire against the knowledge that the new monarch would probably investigate their parent's death. On the other hand, the emotional space between Elizabeth and her heirs, who were all first cousins or more distant relations and had not been raised near her, made it more possible for an heir to get impatient and plot to do something rash. Lady Margaret Clifford was accused of trying to poison her for just this reason.

Then we have the issue of religion. England was thrashing its way through the Reformation: Henry VIII had forced it to break with the Catholic Church, which upset a lot of people; then Mary tried to force it to go back, which upset a lot of other people; Elizabeth's return to Protestantism made those people happy, but she spent much of her reign battling Catholics who viewed her as a heretic, both inside the kingdom and out (i.e., Spain and its armada). She personally didn't want to be succeeded by a Catholic, and a lot of her subjects didn't want that either. This was a major stumbling block for Mary, whose parents had been Catholic and who was raised in the Catholic French court during a time when there was strife and bloodshed between the two camps. One of the things that eventually clinched it for James Stuart was that he was raised Protestant, and could be trusted not to jump on the other side of the see-saw to cause more civil unrest by raising up Catholics and persecuting Anglicans. A Catholic heir would also be even more likely to join a plot against Elizabeth, or to be the unknowing object of one.

A third huge deal was that the succession had gotten very weird over the past several generations. In the long term, of course, there was the question of how much right the Tudors had to the throne, and threats they faced from heirs of the previous dynasty. In just the recent past, Henry VIII had deliberately cut out descendants of Margaret Tudor in his will, stating that his son Edward should be king after him, and if Edward had no heirs and Henry hadn't had another son, Mary or her heirs should inherit, and then Elizabeth or her heirs, and then the descendants of Mary Tudor (Frances and Eleanor) and their issue. Edward took it further by writing out a "Device for the Succession" that laid out who was to follow him:

1. Any male heirs Frances Grey might have

2. Any male heirs Jane Grey might have (later changed to "Jane Grey and her male heirs")

3. Any male heirs Katherine Grey might have

4. Any male heirs Mary Grey might have

5. Any male heirs Margaret Clifford might have

6. Male heirs of their female heirs in the same order

You'll note that he completely cut out not only Margaret Tudor's descendants, but even his older sisters, who he viewed as illegitimate. (Plus, he obviously didn't consider women as potential monarchs. Jane's bit was only changed once it was clear he was dying and Jane was not going to have a baby immediately.) But all of this was questionable: Parliament had given Henry the right to rearrange the succession because of his messed-up marital history, with only one young male heir everyone accepted as legitimate and two female heirs he'd declared bastards on shaky grounds, plus the religious problem. Edward didn't necessarily have the right to designate a successor and cut new people out, but there were clearly people willing to enact his wishes after his death, even if Mary and Elizabeth would throw out his rules. All in all, it wasn't entirely clear what the most appropriate line of succession was and what could and would be accepted.

I suppose the tl;dr of all of this is that Elizabeth did have an heir, really. It was understood that the Greys and Cliffords were in line after her, going by her father's scheme; the Stuart descendants of Margaret Tudor also had a blood claim but weren't necessarily shoo-ins. In pop history and historical fiction, writers focus heavily on the need for a crown to pass directly from father to son and a crisis occurring whenever that's not possible, but that's not really how monarchy worked. It was certainly seen as the best possible option and a sign of favor from God, but it's not like civil war would automatically erupt if it didn't. Henry VIII himself wasn't intent on having a male heir because he thought there would be nobody rule the country if he didn't, but because he was concerned that God was denying him a son in order to show that something was wrong with his marriage. (The Romanovs are another good example of this. Pop culture would have it that there was nobody in line after the hemophiliac Alexei, when actually Nicholas had cousins further down the succession; they simply worried about the optics of the Tsarevitch having a terrible chronic illness.)