Well I'll start with my three favourites and then the rest of my winning team:
Eevee is generally Eevee but I tend to refer to him by several nicknames; Little One, My Blue-eyed Heart, Dear Heart...its a real love bond between us, so they're more endearments than anything.
Charmeleon; Chari because I was nine and the soft growl Charmeleon make sometimes sounds like that.
Flygon; Sango is her name because its a beautiful name, means "coral" and Sango is a very elegant and pretty Flygon.
Typhlosion; Firefriend because he's just that, a friend of fire.
Lapras; Orlith, its a name I got from a book of mine. The namesake was a very assertive gold dragon so I liked it.
Rapidash; Brego, it means "prince" in the LOTR Elvish and I always found Rapidash very noble.
Raichu; Suri, it means "princess" and my Raichu is very spoilt sometimes.
Pidgeot; Dusty, again I was young and I loved the attack "Whirwind".
Scyther; Joss, after my cousin who loves Scyther.
Yeah yeah so I'm a ContestShipper. Sue me.
My Gameboy is actually dead right now so I can't remember the rest.