I always like to name them as if they were a character in the game, it helps me get into the game a bit better. The humans have codenames but the Aliens use their real names and I used a random number generator for Geth. What about you? Also, I'll put all my character names down below for fun and such.
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(*Haven't Unlocked Yet)
Human Male Adept: Twister
Human Female Adept: Tsunami
Asari Adept: Sole Ru'ki
Drell Adept: Korbush
Asari Justicar Adept: Kee'leh Tano
Pheonix Adept: Whiplash
*N7 Fury Adept: Raven
Human Male Soldier: Gallows
Human Female Soldier: Ruby
Krogan Soldier: Urdnot Durge
Turian Soldier: Volavis
BF3 Soldier: Magma
*Batarian Soldier: Slarn
Vorcha Soldier: Flakk
*N7 Destroyer: Kodiak
Human Male Engineer: Gadget
Human Female Engineer: Quark
Quarian Female Engineer: Neeko'Xen vas Tonbay
Salarian Engineer: Petil
*Geth Engineer: Unit 2729-B3-E
Quarian Male Engineer: Lato'Tin vas Moreh
*N7 Demolisher: Eruption
Human Male Sentinel: Hammer
Human Female Sentinel: Gemini
Turian Sentinel: Tulati
Krogan Sentinel: Urdnot Marr
Batarian Sentinel: Thok
*Vorcha Sentinel: Nal
N7 Paladin: Fortress
Human Male Infiltrator: Gilgamesh (named after my Shepard and GT)
Human Female Infiltrator: Swift
Salarian Infiltrator: Seato
Quarian Female Infiltrator: Teeto'Kal vas Jotun
Geth Infiltrator: Unit 5736-S1-I
Quarian Male Infiltrator: Pudi'Mo vas Klah
*N7 Shadow: Katana
Human Male Vanguard: Wreak
Human Female Vanguard: Gorgon
Drell Vanguard: Vezhor
Asari Vanguard: Alera Ton'le
Krogan Vangurad: Urdnot Wramm
Pheonix Vanguard: Wraith
*N7 Slayer: Chariot