Add ham to the bowl of a food processor; pulse until finely chopped but not pasty, 6 or 7 pulses. (You may need to do this in batches.) Transfer to a large bowl; repeat with remaining ham.

Stir mayonnaise, onion, relish, and mustard into chopped ham mixture until well combined; add more mayonnaise if mixture is too dry. Chill ham salad until ready to serve.