I think the biggest challenge is good/safe footing, and the second biggest is a place to clean up after. Outerwear is a thing, but having mud splashed all over the horse and gear and then returning in the pouring rain does add to the disincentives on top of the rest - so a walking trail ride may be a more pleasing choice than a full on regular arena ride.
If you’ve got a warm covered wash stall to return to and a place to hang up all the wet gear to dry that may help make it more attractive.
In Southern California, there aren’t a lot of arenas that are properly engineered to handle a lot of rain, so they tend to be closed at boarding stables when that’s happening.
I usually have alternatives to riding in the rain so I don’t tend to - again footing and cleanup - but I will totally do it if there’s a particular reason to. A long waterproof coat with a split for riding and straps to stay around your leg is a nice addition.