The Versus Seeker is an item in certain Pokémon games that allows the player to re-battle opponents that he has already defeated. It is not obtainable in Pokémon Emerald because it does not appear in the game; however, there is a way to re-battle trainers in the game. The player can call trainers that they've defeated on the Pokénav and they will challenge the player to have a rematch.
Certain trainers give the player their phone number once defeated. This gives the player the ability to call those trainers. The trainers also may call the player themselves. If the trainer calls the player then that usually means they are hoping for a rematch with the trainer.
The player can also have rematches with gym leaders by calling them after the player has successfully defeated the Elite Four. These rematches will be double battles and the gym leaders will be much more powerful than the first time the player battled them.
The Versus Seeker is obtainable in the Pokémon Fire Red, Leaf Green, Diamond, Pearl, and Platinum games. In those games it can be charged by walking and used to see if nearby trainers wish to have rematches. It's battery is recharged when the player walks 100 steps.