Hot water heater leaking

Yes

Right, it won't

Time for a new heater before it gets to be a big leak. Ed

Replace the relief valve. If you still have occasional dripping from the relief, you may have excessive pressure due to thermal expansion:

Jim

It is not difficult to replace the pressure relief valve, but with the increased pressure, that other leak may grow. It sounds about time for a replacement.

I vote with Jim, but since the unit is 12 years old, it may be time for a new heater. You also may want to check the pressure in your home, it may be a little high.

Also wanted to add that those valves are both pressure and temperature sensitive. Just how hot is the water?

If it leaks with the heat turned off, then it is a bad valve. The odds of fixing it by opening it repeatedly is very small.

You can replace the valve cheaply enough, but then you still have a 12 year old heater with a leak. Get rid of it now while it is a small leak rather then in 2 months when it is a major leak, and save the $10 the valve will cost.

That it leaks more with the heater on is probably not a problem; my pressure can temporarily go from 50psi to 80psi when the heater cycles (some day I will get around to installing an expansion tank), but it could be a problem if it is going too high. For $5 get a pressure gauge and check it.

T & P valve costs less than $10.

Any other leak from the tank says the tank has failed and needs to be replaced.

You state that the problem is water comes out of the pressure relief pipe. Thus, the tank isn't leaking the valve is leaking. What is the logical solution getting a new tank or getting a new valve? Right. Buy a new valve for $5.00. Failure of these valves is not uncommon, and they may fail after only 3 or 4 years. BTW, many hot water tanks last much longer than 12 years. The first tank in my house lasted 21 years and the valve started leaking and needed replacement after about 6 years.