From Halopedia, the Halo wiki
Avenge Me is a Bungie.net multiplayer medal available in Halo: Reach. It is earned by being avenged by a teammate. It is represented by a skull with a red cross on its forehead, on top of an eight-pointed blue-and-gold star. This medal will not show up on post-game carnage stats in the game, and can only be viewed on Bungie.net.
On March 31, 2012 Bungie.net ceased updating Halo game stats and player records, transferring all record-keeping responsibilities to Halo Waypoint. As Avenge Me could only be viewed on the site, it is effectively no longer awarded.
This medal is likely named in reference to the 1984 film Red Dawn. The day before he is executed, Tom Eckert yells "Avenge me!" at his sons to urge them to continue their partisan activities against the Communist invaders.