The term “conquer” implies effort, be it in sport or in battle. Within the context of this letter, the effort would be holding fast to what they have received, even in the face of trial. The Lord spurs his people to perseverance—that requires effort!—with the promise of a reward.
12 The one who conquers, I will make him a pillar in the temple of my God. Never shall he go out of it, and I will write on him the name of my God, and the name of the city of my God, the new Jerusalem, which comes down from my God out of heaven, and my own new name.