Offering the paper cup option would be great!
We tried the paper cup option with our Cedar Fair passes this past year and it was much better than the plastic cup.
Here are some observations:
Paper cup (Cedar Fair Parks, Dorney, KI, CP, KD): 15 minutes or so wait time between drinks, cup is on the small side, sometimes there is a wait to process the pass transaction (even at drink stations), you can bring your drink on line for almost any ride, no worrying about misplaced/stolen cups, great for rides with different entrance/exit locations, increased number of trash cans and servicing, increased litter patrol. Consistent policy at all parks visited.
Sports Bottle (Six Flags, SFNE, SFGA, SFA): Refill over and over no problem, greater volume than paper cups, worrying about misplaced/stolen cups, pain in the neck to put in a locker, bees/wasps around cup holders. Consistent policy at all parks visited.
Paper cup vs sports bottle operator view: Inventory and disposal costs, customer satisfaction, sponsorship/marketing opportunities, cost per drink (beverage only vs beverage & cup), control of refill period, trash control throughout the park.