You can trade a Poliwhirl for a Jynx in Gen I (yes I am one of the rare freaks who wants a Jynx on his team), but is it possible to nickname?
(btw Jynx is awesome because it has STAB Ice Punch, a great Special stat and Psychic typing)
EDIT: I'm fully aware of not being able to nickname traded Pokémon, but I'm hoping for some bug in Gen I because the default is LOLA.
No, the only Jynx legitimately available in RBY is nicknamed. ... Notice that I said legitimately. You CAN get a nicknameable Jynx, but only through glitching. To do so, you'll need a Pokemon with Fly, a Pokemon with a Special stat of 72, Pokeballs (obviously), and good timing. This video explains/shows you how to do it: https://www.youtube.com/watch?v=X_mMUZgeSNQ
No, it counts as a trade. And unless you managed to land the same OT, Jynx cannot be renamed. Even though the concept of ID and SID doesn't exist yet, that's still one heck of an odd.
PS: Just because the game is buggy doesn't mean there's a bug for every convenient measure :c
PPS: Jynx is a beast in RBY. You have the right idea putting it on your team.
Hope I helped!