I haven't played stardew valley yet sadly but I was an avid harvest Moon fan and they did this in harvest Moon DS. The name of the person on the rock (idk why but the gravestone was a boulder) is the actual name of your character, unless you put in the gba fomt/mfomt which it'll give the name of your farmer from the gba cartridge. That being said it would be cool if eventually got a real story on this character.
