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This rock thingy lengthens the duration of rain from 5 turns to 8 turns. It can be found on Route 14 and Route 19. I am pretty sure you can find it somewhere else too, but I don't really know.
In battle, if a Pokemon uses Rain Dance while holding a Damp Rock, it will extend Rain Dance for eight turns, instead of the usual five turns. Outside of battle, however, there is no such way. Sorry! D:
Hope I helped. :)