According to Can I Give My Dog...?, no, it is not a good idea to give a dog sleeping pills. Dogs have a different metabolism than humans, and it is not safe to give a dog prescription sleeping pills that were created for humans to use.
Benadryl for Dogs suggests giving a dog a small dose of Benadryl if it has trouble sleeping or is restless. Benadryl is an over the counter medicine that can be safely consumed by canines. The rule of thumb is to give a dog 1 milligram of Benadryl for every 1 pound that it weighs.