A trash bag for a medical office sized trash can over the tiller works. So does a shower cap.
Of course if the water is too deep there is only so much you can do, scooters do not swim.
Of course the rain doesn't usually last more than 30 minutes or so (especially super heavy downpours), so some good choices during rain are longer attractions, such as American Adventure, hall of presidents or go in somewhere like living with the seas where there are indoor things to see and spend some time.
Other choices at Magic Kingdom are Country Bears, Tiki Room (these work well together as there is a minimal uncovered distance between them), Monsters ain't Laugh Floor, Small World, Haunted Mansion (now that they allow
ECVs through the queue and they will be waiting in a covered area when you get off), Philharmonic.
If at Animal Kingdom, it is a great time for the safari, the animals will generally be more active and the scooter gets parked in a covered area.
Eating at Garden Grill if there is availability works, as the scooter gets parked indoors. You can also do living with the land and Soarin.
Spaceship Earth works as well, even just standing under it works due to the drainage system it uses, but it is a longer ride, ao that helps too. Tons of options om EPCOT.
At Hollywood Studios, Star Wars Launch Bay works well. If the line is short or you have lightning lane access, Mickey and Minnie's Runaway Railway can work too. Unfortunately, there are more limited options here. Rise of the Resistance can also be an option. Muppets can sometimes do the trick.
Unfortunately there aren't a lot of other choices now at Hollywood Studios.
In all cases, if you aren't indoors, cover the tiller and follow the rental change advice (if rented) for rain.
If you are trying to exit the park and feel the path is too flooded, ask a cast member for advice.