You don't have to exile each card you find with that name. If you want to leave one in their graveyard, for example, you may.

Unlike most effects of this kind, The Stone Brain's ability allows its controller to choose the name of a basic land. Note that you may only search for up to four copies of that card, even if you find more of them.

You don't have to choose the name of a card that you've already seen this game; you may guess at a card you think your opponent might have in their deck, if you'd like.

The ability has no effect on permanents with the chosen name that are on the battlefield.

You target a player as The Stone Brain's activated ability is put on the stack, but you don't name a card until it resolves. Notably, this means that players don't get to respond to the ability knowing what card you will name.