(Click to enlarge graphs)
As you can see, mythological boys' names were pretty negligible until the mid-1990s, after which they've had quite an explosion, with boys named Phoenix, Odin and Ares leading the pack. Girls having mythological names was more common than boys in the past, but they've increased as well, and the composition of the names has changed dramatically. In 1940, Minerva and Vesta were the most popular (a virgin Greek warrior goddess and a virgin Roman goddess of the hearth ... I'll let you draw your own conclusions from this). Now it's Athena and Isis, unfortunately for those who watch the news from Iraq these days. (Note: an earlier version of the girls' graph omitted the name Athena; thanks to reader John for noticing it.)
Categorizing baby names is not straightforward; there's a judgment call involved. You can cast a broad net, and accept those called Amon, which is coincidentally both an Egyptian God's name and a Hebrew name, which pretty much makes the list meaningless because it's dominated by such names. Or you try to judge through semi-quantitative methods whether a name would, by a reasonable person in American society, be thought of as mythological.
So I limited the mythologies to Greek, Norse, Egyptian and Roman, because they're the most well-known mythologies in this culture. I had to pass on Celtic, because so many of their names are both mythological and common (like Brigid or Dylan). I used nameberry.com's database of name origin's to see whether names were mythological in origin or shared the name with other, more popular, traditions.
The graphs start at 1940 because even though the Social Security Administration publishes them back to 1880, the data is extremely unrepresentative in the early years.
The list of names I started with and which I eliminated at each step are in my GitHub repo, along with an IPython notebook of the code I used to analyze the data and make the graphs.
Gotta catch 'em all... okay, a few of 'em.
It appears that only the boys' name Yadon appeared post-Pokémon; rather surprisingly, Lizardo appears once in 1970 before reappearing once in 2010. This dataset is riddled with errors, however, especially before digitial data entry, so it's quite possibly apocryphal.
Girls named Eevee, Amaura, Kimon and Kameil only appeared post-Pokémon; Abra has been around since the mid-'50s and enjoyed a brief surge around the time the Steve Miller Band's "Abracadabra" was playing on the radio. I remember those days; the lyric "I wanna reach out and grab ya" was pretty racy for Top 40 back then.