If my DM can expect anything from me, it's to make a character out of a pun. There's nothing quite like introducing my character and watching my DM roll their eyes, hearing other players laugh, and knowing this character will be subtly referenced for as long as we live.
Here are a few, enjoy.
Punny character concepts:
A gnome bard that only plays a steady beat on a drum: metrognome
An undead centaur who starts every combat by saying "don't beat the dead horse"
A druid who enjoys lining up small creatures by height aka critter sizing
A bard who only hits on demons and devils, then complains about being fiend-zoned
A werewolf cartographer who only makes maps of the underdark: Lycan Sub Scribe
A centaur necromancer: necroprancer
Aarakocra Cleric: bird of pray
Punny character names:
Auxi Tosin: a bard who is very good at calming people
Indraroom: any loxidon (elephant humanoid)
Xer'ox: any kenku mastermind rogue focused on forgery
Ly, Ka, and Hu Man: 3 kobolds in a trench coat
Claire Voiyant: a on the run halfling fortune teller aka a small medium at large
Have more? Comment them! Maybe they'll make the next update.