How do YOU choose a name for your character?

It depends on the game. I recall having a hard time in Oregon Trail because they were just names, and didn't matter at all unless somebody died. I have trouble when you have just characters you need to assign named for that you know nothing about.
I know that on the school's copy of Oregon Trail kids liked to give inappropriate names so they could have funny tombstones, but I had gotten it as a gift for my own computer a year before the school got a copy.
If the game has a default name, I usually go with that.
On the other hand, on games like Bard's Tale, the Gold Box Games, Might and Magic, or Icewind Dale I might give names to try to help me remember who is what class.
Hence I'll give the fighter a name like "Fighto", the Barbarian the name "Barb", call the Paladin "Pally", the Archer "Archie", and the wizard "Wiz", etc. Otherwise I might instead go with famous names associated with the class like calling a Ranger "Rick" or "Strider" or a Bard "Homer" or "Shakespeare" (and I like to make the latter a spear wielder). I've named a Paladin "Joan" from time to time. I'll use "Conan" for the barbarian, especially if you don't get to pick gender and are creating male character.
I've used both methods in the Bard's Tale series (original) with Pally and Wiz adventuring with Homer and Gandalf.
I've been uncaring enough to do some runs in the Gold Box, Eye of the Beholder, and Infinity Engine games with a Cleric/Fighter/Magic-User named CFM and/or a Fighter/Magic-User/Thief named FMT. Quick, and easy to remember exactly what he or she can do. That's actually how I named my "all multiclass except the Bard" party: "R/C", "F/D", "F/M/T", "C/F/M", "C/T", "Singer", although I swapped "C/T" for a paladin so I'd have all the classes and so somebody could actually hit enemies as I'd started in HoF mode at level 1.
Heck, I did my only playthrough as a Jedi Guardian in KOTR with a character named "Guardian". I say only because I liked using the force powers more rather than a soldier with a few buffs.