A bandana, or a brightly colored length of grosgrain ribbon (will hold up better surprise Florida showers) to mark the handlebars, or a favorite Disney beach towel to cover the seat back...
Gloves are good idea; you don't have to spend a fortune - just some inexpensive golf or biking gloves from
WalMart or Target will do fine.
If you hang a bag(s) off the back, make sure that you don't overload the chair - when your rider stands up, the chair will tump over backwards if it is overloaded. Also don't hang anything off the back that you won't take with you on the rides; Disney is a safe place, but there's only so much temptation some folks can stand...
If you need to carry a lot of stuff, you can always rent a locker up front.
If your person will be wearing shorts, remember to put some sunscreen on the tops of their feet, and their thighs, as well as their shoulders and arms, and the tops of their ears (if exposed). When you sit all day, those are the areas that can easily get over-exposed to the sun.
Probably the thing you will wish you had the most, and we all always forget about is a cupholder - but finding one you like is tricky, and can get expensive - so instead, consider a water bottle lanyard (they sell them at WDW now) that has a soft, rubber holder than goes around the neck of the water bottle, and a lanyard. You can get ice water for free at any QS to refill your bottle all day.