I've been using 'Shadria' for a long time. It doesn't help that I have started a 'story' where the trainer is the same person travelling around. Aka, the trainer in every game IS the same person. So, same name. Have been doing this since Soul Silver.
I'm probably going to use my go-to, which is just "Hunter". It works well for me, Hunter is one of my Trainer OCs that travels around the different regions, so it's basically the same person every time XD
I normally just go with whatever the default name for the character is, but I might go with Kai this time. It was Sun's name in the code for S/M and I like the name.
The first time I play each game, I try and find a name that I think fits the protagonist's appearance. Like, I always thought the girl from Black/White looked like a Steph for some reason. Though, I usually name my character after myself or a similar variant (e.g. JJ, JoJo, Jo, Sephine, etc). If I'm playing a challenge, I tend to name my trainer after the challenge (e.g. Belladonna for a poison monotype challenge, Zero for a Nuzlocke, etc). Though most of the time, I spend a minute staring blankly at the screen, trying to think up a name before calling the player something I end up wishing I could change. I was really thankful that there was an option to change the name of your character in Pokémon Super Mystery Dungeon. Sometimes, I try and have some fun and call me character something like 'You'. It's always fun to read something like "You found a Potion!" and "I'll show you the true strength of Pokémon, You!"
I usually name the player character after me, Sam. It makes me feel more involved in the story and the game more fun~
I typically pick whatever the default name is for the particular character I'm playing (Red/Blue for the Gen1 chars, Ethan/Lyra for HGSS, Brendan/May for RSE, and etc.), but sometimes I choose Miroku, especially if I'm doing a Nuzlocke run.
I'll probably use Havok. Been doing that since Gen IV and after I took to writing a story about the character namesake in Black and White. Other one I'll probably use the default for whichever gender I choose
I've always named the character after myself. It just makes me feel like part of the story. Although if I could replay X without deleting my data (like you could in previous games), I would try playing as a boy and probably give him some Japanese name, lol.