I came home today and found that the gas water heater (traditional tank style) was not working. The pilot light was out so I relit it and the heater came on. After about 10 minutes I noticed that it had shut off. There was a sound like water dripping on a hot surface coming from inside the tank, maybe a drop every few seconds. The pilot light was once again out. There is no sign of water on the floor. Time to replace it? The unit is a Whirlpool, about 10 years old.
In then end it was indeed the thermocouple. At some point the water heater shut off and things got cold enough that condensation formed when it was restarted. Since replacing the thermocouple, the problem has not reoccurred.
Leaks in a hot water heater don't get better and are not generally repairable, so it's new water heater time since it's leaking (and it sounds like it is, enough to put the flame out.)
I would suggest shutting both the water and gas valves for the hot water heater off to head off any excitement if it decides to get worse than "1 drip every few seconds." Depending on your family's tolerance for cold water and your plumber's "get it done now!" rate a few nights at a motel might be cheaper than a quick fix from the plumber (or you may have other options such as showers at a gym, pool, etc.) - if DIY-ing it, it buys a little time for shopping around instead of taking the first HWH you can lay your hands on.