I usually use whatever the actual trainer's name is.
For example, in Diamond, I was Dawn. In Emerald, I was May.
I do occasionally use my own name though.
Current name I've been using is Shnell, all games Gen 3 and so on.
Other names used:
Matt (my real name, used it when I first got Pokemon, gen 1-3)
Kazuto (I don't know, I just liked the name...used in one of my Heartgold runs, gen 4 only)
Veriday (used in PokeMMO for a bit before changing it, gen 1 + 3)
Alpha Sapphire- Nate (This is a name that I resort to if another character in the game uses my real name)
Omega Ruby- Nate (Haven't created the file yet, but this will be his name)
X/Y was the first pokemon game I could play where my real name would fit (Michelle), so I took advantage of that. I don't regret it thanks to the fact that you can customize your character in that one. She looks like me now, so it's great.
Before X/Y, I would usually use Mish since it's my nickname.
Alpha Sapphire has been the first one where I chose a different name for my character. I chose Maren because idek it's one of my favourite names.
I use my own name in all the games I own. The ones I got from my sister use her name, though. I don't really mind / care to reset them since I had been playing the since I was much younger.
My trainer name is always Jessie, which is my actual name. I don't think I've ever used another name in the 15 years I've been playing now I come to think of it, I guess I'm a bit boring
I use my name in X and Anna's my name in the OmegaRuby (which I'm currently playing and is in no way in reference to May's ORAS demo name but something else). As for AlphaSapphire, I have no idea yet. Still debating on that.
I named my trainer Risu, which means 'squirrel' in Japanese. It's funny because my girl names are usually random and unrelated to my real name, but if I play as a guy, his name always starts with the letter of my first name: K. This is the second time I played as a female trainer in Pokemon. No regrets, but that's certainly going to change in future games. I guess I share a stronger affinity with male protagonists. Not sure why, haha >u>
Trent like my real name, B2 and X before that. Alpha Sapphire is Rosemary, was going to go with May but my brother did so I decided to go with something a little different.
Kayla, which is my middle name. I use it for all of my characters/games.
I like it better than my first name which I don't use for anything except school x)