The item you are describing is the Verse Seeker which does allow the player to re-battle trainers they have already challenged within the game. The only games that have this device would be FireRed/LeafGreen and Diamond/Pearl/Platinum. Other games, such as Emerald, HeartGold, and SoulSilver are dependent upon the game's phone which allows you or the other trainer to call for a rematch. As for Ruby and Sapphire, you must wait until one is ready to have a rematch using the PokéNav.