Which version is grammatically correct?

Hey. I want to add text to game to choose level and how to say it grammatically correct in English?

level select or select level. or should I use word "stage" or "map" or what?

Both are fine. Having the verb first would be an instruction or command, and saying "level select" would be the seen as the name of the screen. Whether to use level, stage or map is up to you.