Usually I just give them my name (if female) or canon names. Unless I'm feeling childish that day.
In FFVII I renamed Barret "Douche", most other characters kept their canon names (including Cloud).
Tidus in FFX was renamed Airhead for no reason.
Marche in FF Tactics Advanced was once named New Kid, just so he could say "My name isn't New Kid! Its New Kid!"
In Pokèmon my main character is almost always named Usagi (except in Black, where its actually male and named Conejo, both mean rabbit in two different languages) and the rival something random or their canon name. I may even have named him Roxas one time. The Pokèmon's name range from normal, to the outright ridiculous. In SoulSilver half of my party got Warcraft-names (Sylvanas the Totodile, Krasus the Charizard and Thrall the Torterra.) When I started breeding Snivys I ended up naming all of them after Slytherin-members (suitably enough my starter's name was Salazar) seeing as I ended up running out of names some were named after the same character, as such I had three called Lucious, Draco and Malfoy, and two named Severus and Snape.
In Roller Coaster Tycoon 2 my friends and I once had fun with giving guests silly names and then killing them/dropping them into the wild, only to get messages such as "A little fish has drowned" or "a man with a map is lost and can't find the park exit".