Even Tigole thought it was a "cool idea," so there is the ghost of a chance that we will see it in a Wrath content patch. I would definitely enjoy the chance to make a quick pass at a boss or two with four of my friends, and get some extra badges for it at the end of the week. It would also be really fun to get to face some old bosses again, especially given Blizzard's steadfast refusal to implement heroic Deadmines.
The idea is as follows:
PvE arena points are earned just like PvP arena points, and can be spent on rewards in a similar fashion. An alternate suggestion (also by the OP) is to reward badges based on PvE arena rating, which I think is a better idea, as well as being easier to implement, because it ties into the existing badge system.
Round up four of your friends and form a party.
Talk to the PvE arena master and get dropped into a cage match with a random dungeon boss from classic or BC, tuned up to level 80.
If you win (that is, the boss dies before the last member of your party), your PvE arena rating increases. If you lose, it decreases. Bosses have their own rating, which develops week-by-week similar to players', so beating the harder bosses will do better for your score, and they'll try to match your team up against bosses that are an appropriate level of difficulty for you.