Place diced potato in a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, 8 to 10 minutes. Drain.

Heat oil in a large skillet over medium heat. Add cooked potato and brown on all sides for about 3 minutes. Remove from the skillet and set aside.

Add ground beef to the skillet; cook and stir until browned, crumbly, and no longer pink in the center, 4 to 5 minutes. Stir in bell pepper, onion, and garlic; cook until softened, 3 to 4 minutes. Drain as much fat from the pan as possible.

Stir in tomato sauce, water, olive brine, oregano, salt, and pepper until well combined. Add olives, raisins, and reserved potatoes; stir well. Reduce heat to low and simmer picadillo for 8 minutes, stirring occasionally.

Heat a nonstick skillet over medium heat and spray with cooking spray. Place a tortilla in the hot skillet and cook until browned, 1 to 2 minutes per side.

Spread about 1/4 cup hot picadillo on one half of the hot tortilla. Top with Mexican cheese, then fold tortilla in half, pressing down with a spatula. Cook until cheese is melted, about 1 more minute. Repeat to cook remaining quesadillas.