How do people usually give nicknames?

I've been chatting with my friend for a long time now so it got to the point when teasing (or making fun of) each other is really normal.

So just recently, I just called her an evil troll jokingly (because she was trolling me and she's really savage while being at it), sometimes she'd call me the same thing back but this time she asked me to give her a cuter name.

I couldn't think of any at that point in time so I just replied the current one she has is already cute (which is true). But I did tell her I'll get back to her once I have something in mind and it's kinda hard to find one that suits her.

Hence, I'm wondering how do Japanese usually give their friends nicknames? Any idea would be appreciated ;-;

There are no hard rules for creating a nickname. A lot of (but not all) Japanese nicknames tend to be abbreviation of or derived from their full name, like shortening the first name or last name, or truncating the last name and first name. They do not have to mean anything, as long as it sounds snappy (short and easy to pronounce).
Examples: Morita Kenichi became Moriken; Wada Akiko became Akko; Kato Ayako became Katopan; etc.

I'm a Japanese resident who went to school in the U.S., and I don't feel that there are any differences in the way people give nicknames.

Actually, I'm not sure if I understand the intention of your question, because I'm sure she's asking you to give her a nickname based on your rules.

I tend to think the nickname will come from others instead of oneself.

Just go for Trokko, makes the whole troll thing sound cuter, doesn't it?