Yes they do
Source: I can't find my link (guess who cleared his history) but everyone I've talked to (who have used rocks) have said that weather is elongated by it.
EDIT: Yeah that wasn't a good source and I have a better one now. The weather rocks are extending the length of weather on Custom Games Gen 6 for Showdown. HOWEVER, none of the main Pokemon sites like Bulbapedia, Serebii, or the DB have said that the weather rocks extend weather.
I also talked with someone on showdown who played in-game with weather and he said that he used Damp Rock to extend drizzle's rain from 5 to 8 turns.