No, homeowners policies don't cover holes made in your yard by the draining of a neighbor's backyard pool or pond. But there is some good news. Home insurance policies do include personal liability coverage for injuries or property damage to others. It's not your policy that will pay the claim; rather, it's your neighbor's homeowners liability coverage that should pay the claim.
You will need to document your claim. Ideally, you'd do this with pictures, a statement from a contractor certifying that the damage was caused by your neighbor's actions, and an estimate of how much it will take to fix the problem.
Once you have all the documentation to make your case, ask your neighbor to file a claim under his homeowners policy. In almost every case, home insurance companies do not raise rates for liability claims.
One cost that the neighbor probably will incur out of pocket is the cost to build some kind of barrier to prevent the problem from happening again. It's something your neighbor should have done when the pool and the pond were first put in.
If your neighbor refuses to file a claim with his insurance company, or if his insurance company won't pay, the next step would be to file suit against him in small claims court. Bring your documentation. If you win -- and you should win -- your neighbor's home insurance company will probably end up paying your damage for the backyard.
Best of luck.