Place the ham, egg, celery, and onion in a food processor.
Use the pulse feature in short bursts, about 15-20 times to mince - not puree - the ham and other ingredients. Scrape down the sides of the food processor every so often to make sure that everything is combined and minced.
Combine the ingredients for the mayo dressing in a medium-sized bowl, taste-testing and adding more ingredients to taste. I prefer a bit more relish in mine!
Scrape the ham mixture into the mayo dressing and combine thoroughly. Cover with plastic wrap and refrigerate for at least a few hours before serving, but this is best if left overnight for the flavors to meld together.