I decided to humor Cindy by having her service the Regalia for a change. To my dismay, I paid her a ton of Gil and saw no change in performance or appearance. What exactly did I pay her to do?
It repairs the Regalia as well as cleaning any mud off of it!
However, it's worth noting that you can upgrade the Regalia to be more durable by completing Cindy's missions and getting the fiberglass coating.
Later in the game in Altissia, you can purchase the Superhydrophobic coating (which does the same, only better) for 20,000 gil.