Place eggs in a saucepan; cover with water. Bring to a boil, remove from heat, and let eggs stand in hot water for 15 minutes. Remove eggs from hot water, cool under cold running water, and peel.
Cut each egg in half lengthwise; place egg yolks in a bowl. Mash yolks with a fork.
Stir mayonnaise, feta cheese, basil, tomato, black olives, and balsamic vinegar together in a bowl; stir in mashed yolks. Spoon yolk mixture into a resealable plastic bag; snip off one corner of the bag.
Place egg whites cut-side up on a serving platter. Pipe yolk mixture into egg whites. Season deviled eggs with black pepper; garnish with additional chopped basil and crumbled feta cheese.
If you have any leftover yolk mixture, save it and make an egg sandwich with it later, or cut open an avocado and place the egg mixture in the spot where the seed is and enjoy!
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