What equivalent series resistance (ESR) capacitor should I replace bad one with?
A value of 1 Farad hints at a clock or memory backup capacitor. This will power a chip drawing current in the micro-amps range. Therefore, high ESR will not matter at all.

ESR matters for critical decoupling, regulator stability, and high currents, none of which are of concern in this application. Since you will likely pay more in postage than in the capacitor itself though, I suggest selecting a higher quality model though.

Your criteria should be:


Make sure the package fits into the PCB footprint
Don't forget about height! Component height can bite when closing the box after doing repairs.
Leakage current, which will discharge your cap. Get a cap specified for memory backup, so it keeps its charge for a long time.
Lifetime in hours at specified temperature (higher is always good if it's affordable)
Select a lower ESR model if it makes you feel warm and fuzzy, but really, it wont matter. 0.1 or 10 ohms ESR will not matter at all, if the chip it powers draws microamps, the ESR voltage drop will be negligible.