Arena NEEDS a match history that at minimum records matches played, opponents, and wins/losses. I've searched for other threads about this, replied, upvoted, and am now posting my own suggestion.
I just played an amazing match, where the opponent made a play just to point out something that I'd missed before beating them. I want to send a friend request to them to say "that was awesome, thanks!", and maybe play them again - but there's no way for me to find their full ID#.
I agree with this suggestion.
The history can include items like deck used, match length, outcome, "Send Friend Request" button, "Rematch" button, "Report" button, etc.
This game, outside of Arena, has such a warm and welcoming community, if we expanded that to Arena, I'm sure a lot more players would stick with the game.
However, I would highly advise against personal messages to people that are not on your friends list. For the players that like to utilize "hate" decks, I feel they would be targeting for some unpleasant messages.