Elizabeth's focus on her virginity had little to do with sex and purity. Ultimately, it served two functions: it prevented a foreign power from gaining too much control of England and it allowed her to maintain control over her male courtiers all vying for her attention.

Marriage and therefore sex would have effectively annexed England to one of the main European powers. Her advisers and subjects were especially terrified of a Spanish match. Her predecessor and half-sister Mary I had married the king of Spain and Philip became, by default, king of England. Mary allowed Philip to exert almost total control over her and England because she believed she had taken a vow to obey her husband. Most everybody believed that wives should obey husbands, of course, but what did that mean for a small Protestant country that wanted to maintain its independence?

Don't forget the significance of religion in all of this. The Anglican Church was still very new and Elizabeth's courtiers, advisers, and subjects were all divided; many were still Catholics. Mary I's marriage to a staunch Catholic had led to the persecution and deaths of hundreds of Protestants. Elizabeth herself was a devout Protestant but took a very moderate approach to religious reforms, preserving a lot of Catholic bells and whistles like clerical dress. This ended up angering both Protestants who felt she was pandering to papists and Catholics who thought she was pandering to heretics but it ultimately allowed her to play both sides and avoid definitively taking one side or the other.

She played a similar game in terms of marriage. There were early talks of a Dutch marriage, to a fellow Protestant. Many of her closest advisers, however, favored a French (Catholic) marriage, because a French alliance would be further protection against the Spanish. But Elizabeth didn't want to commit to anything if it meant alienating another possibility. Despite her much-lauded virginity, Elizabeth entered into marriage negotiations several times throughout her life, all the way into her mid to late 30's.

So marriage would have forced Elizabeth to definitively choose a side in religious and political debates among the European powers. She didn't want to do that. She also didn't want to choose between her various advisers and nobles. She had to maintain a delicate power balance among courtiers who were both Catholic and Protestant, who all put forward various candidates for her marriage. Pretending to entertain all of them but ultimately never committing allowed Elizabeth to adjust the power balance over time. If courtiers were angered about her choice of favorites, she could shower them with titles and land and pursue their favored alliances and marriage proposals for a few years until it was time to pander to some other family or group of nobles.

In addition to the political strategy behind her virginity, Elizabeth was also a big believer in courtly love. She consistently kept dozens of men around her, all pretending to vie for her affections. In part this was because it allowed her to control rogue courtiers who might be scheming against her. If they all had to pretend to pay her court, she could keep an eye on them and adjust her tactics accordingly.

However, Elizabeth also craved attention. She was extremely jealous of her favorites and often asked them to remain unmarried, presumably so she could dangle the opportunity to marry her over their heads for several years to maintain her power. The queen took it very personally when one of her favorites took a wife, often forbidding their wives to come to court and sometimes banishing the men from court as well.

In the case of Sir Walter Raleigh, he married his wife secretly in 1591. But, of course, Elizabeth eventually found out and imprisoned both of them in the Tower of London. After she released them, they refused to apologize and fell out of favor for several years.

At the time of his expeditions in the 1580's, however, Raleigh was still one of the queen's favorites. Because she gave him patronage and charters, it made sense that he would name the colony after her. But why not just Elizabeth?

It's quite likely that Elizabeth actually selected the name herself, as a coded message to her courtiers, her people, and the foreign powers also colonizing the New World. In 1578, Elizabeth had entered marriage negotiations with the Duke of Alençon. Many of her advisers were pushing her to marry and produce an heir, fearing that the kingdom would fall into anarchy upon her death without a successor. However, most of the common people did not support this choice, and a large stream of public tributes to the queen's virginity attested to this opinion.

By the time the queen granted Sir Walter Raleigh his charter in 1584, she had finally made the decision to never marry. The days of playing both sides were over, and she was becoming too old to be a suitable match for anyone anyway. Rather ironically, it was only as she got older that she began truly escalating the cult of virginity and saying that she was really married to her country and all those famous quotes. At that point, it was too late to do anything else.

A lot of people see Elizabeth as a determined and resolute queen, but in many ways she was indecisive and impulsive. She was in a precarious position that required a lot of waffling and playing both sides to maintain power. Her virginity was ultimately a product of her indecisiveness and unwillingness to commit to a marriage. In the early years of her reign, she was considered a virgin because she hadn't chosen a husband yet. She negotiated multiple matches and was perfectly willing to entertain the notion of marriage and sex in theory, but she was too cautious to commit to any one person, religious affiliation, or country. Her virginity then became a sign of her indecision and, to some, irresponsibility. It was only in the 1570's and 1580's that she fashioned her own virginity into a deliberate choice that glorified God, herself, and her people. The naming of Virginia coincided with this period of the escalation of the cult of virginity.

Sources:

A. Somerset, Elizabeth I (New York: Knopf, 1991).

A. Weir, The Life of Elizabeth I (London: Pimlico, 1999).

A. Whitelock, The Queen's Bed: An Intimate History of Elizabeth's Court (New York: Farrar, Straus and Giroux, 2014).

A follow up question if I may: do we know with reasonable confidence that Elizabeth died a virgin? Were there any credible contemporary claims from people who say they slept with Queen Elizabeth?