It seems that the gun had a bad oil finish, which didn't stand the rain. Polish it with fine sand paper and wet it with a wet cloth. Work over it with fine sand paper again. Unscrew the buttplate. Then take some linseed oil, a brush and a hot air gun. Warm the wood without burning it, and apply the oil with the brush. Go on so until the wood is soaked, and end with wiping off any extensive oil. Let the stock rest a couple of day, and then give it the same overhaul with oil. Finally, put the butt plate back and polish the surface with fine steel wool and linseed oil.
The hot air gun speeds up the work with applying oil. A correct, traditional oiling will need some months until the stock is satied with oil, but this method makes it a lot faster. However, I would just use it on a working gun, where aesthetics are not primary, but function is. I also think that the air gun will cure the application of gun oil.
Pete