If the pond is fresh water and the rain is fresh water, the density should not change. If the rain and pond are very different temperatures, I guess the density would change a bit, but the effect is really not very noticeable. However, if rain water runs off into the ocean, it is less dense and it "floats" on top of the salt water. This can cause significant local effects in coastal places where is rains a lot (like the west coast of New Zealand).

Most ponds are fresh water, and rainfall is fresh water, so they would have the same density.[What is the density of fresh water?] Little salty ponds or puddles near the ocean might get less dense with a big rainstorm. Actually, I think the rain storm would probably wash out the sand between the little pond and the ocean. You could think about rainfall vs the average depth of a salty [or "brackish"] pond. If there was an inch of rainfall and the pond was 6 ft deep on the average, how much would the salt concentration in the pond change?

I imagine that rain does change the average density of a pond by some amount as rain water is probably more pure then pond water. I'm not sure at what point the change would be measurable though. Also, a lot more rain water will get into a pond from run-off then from directly falling into the pond, but run-off would also tend to carry some dirt with it as it runs into the pond.