We have replaced the hot water heater thinking that this was the problem, but we are still without hot water in the AM. When we go to take a shower in the AM it runs luke warm. After a while, or after someone takes a shower - in luke warm water, we then get really hot water. We do have hard water, so I was thinking that could be a factor. Suggestions?

Does this only happen in this faucet? What brand is the shower faucet? How old? Is this a recent problem? Have you made any alterations or repairs on this faucet?

No, this happens in the other shower too. My husband has resorted to getting up in the middle of the night to get the shower running - sort of like "priming the pump". He then has hot water at 5:30 AM when he gets up. We have not replaced any of the fixtures, only the hot water heater, and it has been happening for a period of at least 5 years. It is a Moen faucet and was here when we bought the house 10 years ago.

You may simply have to adjust the shower cartridge. Does it turn on from right to left or does it pull out to turn on?
Or
Call moen at 1-800-321-6636 and request a replacement cartridge for the offending faucet/s. it is likely the problem is in the cartridge. Sometimes moen will give you a free one, and for sure it will be genuine moen parts. Moen clone parts are available but some are best avoided. You may have a clone cartridge.