Okay, this one may be a bit weird. We have two full baths in our house, upstairs and downstairs, and a tank less hot water heater ~7-9 years old. The upstairs shower works fine. Downstairs, however, the hot water lasts about 3 minutes then runs cold in the shower. However, if I simply turn it off and back on again, we get another roughly 3 minutes of water. What is going on? Hot water elsewhere stays on as usual, although everywhere there is some fluctuation. The house pipes are old, galvanized steel for many of them. Thank you!
I think what we have here is possibly a poorly functioning hot water heater that is not providing you with a more consistent stream of hot water. As you said, there are fluctuations in the hot water levels which would suggest that the source of the hot water is not properly heating and maintaining a sufficient hot water supply. You can check the temperature setting and also monitor closely in the days ahead the overall performance of your water heater. More about that later.
Another cause (though less likely in my opinion) of your intermittent hot water problem could be related to a partial obstruction in your hot water pipe leading to your downstairs shower. Another causal element may very well be pressure related, though I suspect that is not the case. It sounds more likely you have a either a hot water heater that is not moving sufficient quantities of hot water to you downstairs areas or a clogged pipe.
One economical way of ascertaining which path you may take in correcting the problem is to wait to see if the intermittency of the hot water distribution throughout your house grows worse. If that is the case, then it may be time to purchase a new hot water heater.