we mostly camp in the south west, so we'd never go if the rain put us off! i think you really need to be well prepared though.
everyone needs good waterproofs/footwear and short trousers/leggings so any bits that do get wet dry off quickly. lots of layers cos once you get wet you get cold and cold=camping disaster.
you need to have the right tent too, with space for airing/drying wet gear. this time i'm taking a couple of lengths of garden cane that i can tie to the frame poles to hang stuff off. i seem to have banged on a lot lately about drying stuff in the car during the day, but it works! plus a reasonable living area and entertainments (books, jigsaws, craft kits) for wet mornings/evenings (would never let the rain put us off going out in the day though, that would be grim). also having provision for somewhere reasonably weather-proof to cook.
paying attention to where/how you pitch can stop you waking up in a puddle, or guys coming loose and fly touching inner. choose sites within reach of wet-weather activities too.
when we packed up in a gale by the beach last time it was a complete godsend to have a portable dvd player so we could just bung the kids into the car with that while we packed up unpestered outside (in, as dp remarked, a scene reminiscent of moby dick ).
if i think of any other wet weather tips will post them. for us, it's enough of a novelty just for the kids to have space to run around (last time out they made a "water slide" out of a groundsheet) that the weather really is a minor issue.