Is it normal for my hand to swell after my rotator cuff surgery?

Yes-quickly subsides. The combination of internal tissue pressure from arthroscopy fluid used at surgery and immobilization in a brace can lead to temporary postoperative swelling that includes the hand. The swelling from the arthroscopy fluid usually subsides in hours. The residual swelling from immobilization in a brace should continue to diminish steadily within days. Elbow, wrist and hand motion helps (if allowed).
Yes. Swelling in the hand after any surgery on the shoulder, elbow, or wrist is usually not a problem. The combination of increased blood flow to the arm as part of the healing response along with decreases in range of motion after surgery contribute to the swelling. The swelling usually resolves after days to weeks depending on the individual patient.