"The Bourne Identity" is a pun since he has amnesia and seeks who he is (his born identity). Unfortunately, they degraded the theme in subsequent movies, with only "Bourne Supremacy" having any actual double-meaning, and the latest film is just "Jason Bourne", so they just stopped trying to keep them even thematically consistent.
Asterix is a western comic series full of pun names, so are Piers Anthony's Xanth novels.
Hajime no Ippo is a good pun title. Hajime=beginning, no=of (but works right-to-left), Ippo = step. So the literal title is "The first step", meaning the character starting out, but since Ippo is also the main character's name it can equally literally mean "Ippo's beginning". Ippo also sounds like Ippatsu, which is a root word in phrases meaning things like "punch", "win by a hair's breath", "turn things around with a sudden strike" etc, all of which is relevent to Hajime no Ippo.
http://www.romajidesu.com/dictionary/meaning-of-%E3%81%84%E3%81%A3%E3%81%B1%E3%81%A4.htmlKuroko's Basketball is also jam packed with puns / double meanings in names, related to hair color and other light-based pun names (shadow, mirror, sun, moon, shade etc)
I also watched some shorts of a slice of life anime called
Yurumates. It was fairly funny. Mentioning it here because the title has 3 meanings in it. It's about an apartment building full of useless lazy people (who are supposed to be studying to get into college). Yuru = lazy/easygoing. So it could mean "lazy friends(mates)", but the ru from yuru combines with "mates" to make "room-mates" in English as well, and the main character is also named Yurume.