I would love to see how old my save is for example, for the sake of speedrunning. Other stats could include how many fights you've participated in, which types you've fought most in etc, how much money earned total etc.
There could be a lot of value in having detailed stats in the Arena. I'd like to see an end of match screen where basic stats are shown (ie: Hits landed, blows parried, damage dealt, ect...) and in the barracks a way to get detailed record stats for each character you ever had.
Super extra bonus points if there's a leaderboard integration of some form