The dripping could be caused by a simple bladder infection, that's fairly inexpensive to check.
Does the kitten drip at any other time or only when sleeping? I only ask this because it could be a sign that the cat is incontinent, I should get her checked out by a vet. I had a cat who became incontinent following cathetrisation during an operation but this was due to the irritation caused and it did get better eventually. Sadly the original problem (calcium deposits in the bladder and urinary tract) came back and nothing more could be done for him.
I doubt if it would be a calcium problem at such a young age, it may be developmental but the only way to really know is to ask a vet.