Step 1: Note where the pain is.
The pain sits on the outside of the elbow and gets worse with gripping tasks like swinging a hammer or wringing a sponge. Pain right at that spot, worsened by grip, is the hallmark of tennis elbow.
Step 2: Recall which tendons attach there.
The lateral epicondyle is the shared origin of the wrist extensors. Extensor carpi radialis brevis is the classic culprit, working alongside extensor carpi radialis longus, since both must contract hard to hold the wrist steady in slight extension every time the hand grips something firmly.
Step 3: Eliminate muscles from the wrong side or wrong job.
Biceps and supinator sit in front and handle flexion and supination, not gripping stability. Flexor digitorum superficialis comes from the medial epicondyle, the opposite side, giving medial pain instead. Triceps and anconeus are posterior extensors of the elbow, unrelated to lateral pain from gripping.
Step 4: Conclude.
\[ \boxed{\text{Extensor carpi radialis longus and brevis}} \]