I'll consider the characters and pronunciation of the default name to see if it sounds good before deciding whether to change it.
When playing games, I prioritize immersion (not absolute) and prefer names I like to help me blend into the game's atmosphere.
If I were to change my name, I'd choose an English name, regardless of whether the game is set in Japan, the West, modern times, ancient times, the future, or an alternate world; it won't feel out of place.