Leather can and should be treated, if you do this there is no reason you can't ride in the rain. The same goes for Oilskin although not quite as durable. Marine grade Vinyl is a standard type of cover, and is durable and will last if treated. Any of these options require maintenance if you wish them to last, regular cleaning and appropriate treatment. A lot of problems with seats is caused by inferior thread which can't take the weather and subsequently rots. Hope this helps.
Regards Martin.