Sometimes, a name isn't exactly meaningful or hiding something, but there's still something about it that is weird. And then you read it really fast, and go, "Oh!" and have your chuckle over the pun.
Pun names are just that: Names that make puns. They don't have to be Meaningful Names, but often are. An overlap would be a baseball player named Homer (of which there have been several actual Major League Baseball players with that moniker, although a few were pitchers).
The pun name is a staple of the prank caller, who will disguise it just right so the other guy doesn't get the pun until he blurts it out.
A subtrope is Sir Verb-a-Lot. Another subtrope, for TV or radio stations with punny names, is WPUN. Camp Wackyname is also a subtrope, and "Miss X" Pun is one when used for names. If the name is a Double Entendre, then it's a Euphemistic Name. If a lot of characters in a given work have a punny name, it could be an example of Punny Theme Naming. If a group of people's last names forms a pun, it could relate to The Law Firm of Pun, Pun, and Wordplay. Also overlaps often with Real Joke Name, where someone laughs at what they think is a pun only to be told that that's actually the character's real name. (And they've Never Heard That One Before.)
See also: Who's on First? and Who Names Their Kid "Dude"? For a list of tropes with Punny Names, see Just for Pun.
If it's a title, then it's Pun-Based Title. If it's the name of an entire species designed around the pun, see Pun-Based Creature.
