Toss together arepa flour, cheese, and 1/8 teaspoon salt in a bowl, then stir in water until incorporated. Let stand until enough water is absorbed for a soft dough to form, 1 to 2 minutes (dough will continue to stiffen).
Form 3 level tablespoons dough into 1 ball and flatten between your palms, gently pressing to form a 1/4-inch-thick patty (2 1/2 to 2 3/4 inches wide), then gently press around side to eliminate cracks. Transfer to a wax-paper-lined surface. Form more disks with remaining dough in same manner, transferring to wax-paper-lined surface.
Heat oil in a large nonstick or cast-iron skillet over medium heat until it shimmers, then fry arepas in 2 batches, turning over once, until deep golden in patches, 8 to 10 minutes total per batch. Drain on paper towels.