Perhaps no Game of Thrones character is as beloved as Daenerys Targaryen (played by Emilia Clarke), the daughter of a slain king who has spent most of the series to date amassing forces to reclaim the lands that were once her father's. Also she has dragons, which people seem to like.
So it's not too surprising that fans of the show would name their kids after her. According to data from the Social Security Administration, were 21 newborns in 2012 named "Daenerys," which was never used enough in previous years to show up in official counts (for privacy reasons, the SSA only releases numbers for names used five or more times in a given year).
But wee baby Daeneryses were dramatically outnumbered by newborns named "Khaleesi" — the title Targaryen earned when she married Dothraki leader (or "Khal") Drogo. 146 "Khaleesi"s were born in 2012, making it more popular as a full name than "Betsy" or "Nadine":
In 2010, less than five newborns were named Khaleesi. That rose to 28 in 2011 and then more than quintupled to 146 in 2012.
Another fan favorite character, the itinerant orphan Arya Stark, also seems to have inspired a few hundred baby names. Causality is harder to determine here than with "Khaleesi," as "Arya" is a real name (mostly for boys) whereas "Khaleesi" is a made-up term invented by George R.R. Martin (who wrote the books on which the show is based). So it's possible that "Arya"'s fluctuations in popularity can be attributed to non-Game of Thrones factors.
But the name surged in popularity with girls in 2011 and 2012, after the show premiered. Also suggestive is the fact that it was barely used as a girl's name before Martin's first book came out in 1996 (it was used less than five times and thus not in the SSA's data in 1995, and in the two years previous), and has since steadily risen in popularity:
Indeed, since the book came out the gendering of "Arya" as a name has completely flipped:
Of course, Game of Thrones is hardly alone among sci-fi and fantasy literature in changing parental naming practices. "Hermione," "Draco," and "Sirius" all gained popularity after being featured in the Harry Potter books; the characters Hermione and Draco were present from the first book (out in 1997, movie in 2001), while Sirius was introduced with the third (out in 1999, movie in 2004):
The Hunger Games managed to make the name "Katniss" a thing. It doesn't show up in SSA data until 2012, when it overtook "Maude" and "Lucianne." That was the same year the movie version — in which Jennifer Lawrence played Katniss Everdeen, who must fight for her region in the titular tourney — came out: