I'm just curious, how did these save batteries function?
If I'm not misunderstanding you, then they work pretty easily (and they do eventually fail for all the usual reasons that batteries eventually fail.) The issue relates to "volatile" and "non-volatile" memory. Non-volatile memory doesn't forget things when the power is turned off. Volatile memory does forget everything, so that when the power is returned its values are "random" or else all of the same value. Programs are usually placed into non-volatile memory, so that they are not forgotten when the power is turned off. However, most non-volatile memory has a "cost" to it -- it is hard (or impossible) to modify. (Some is easy, though, such as FERAM/FRAM -- but that is fairly modern and not readily available or cheap to buy.)
To save a game, you need memory you can modify easily. But you also want the game to be recoverable after the power is turned off and then back on, too. Which means it must be non-volatile memory. Before FERAM/FRAM existed, probably the cheapest way to do this was to provide a small bit of low-power SRAM (static RAM can be very low power, though not necessarily, as it depends on a lot of factors) that is "battery-backed" so that it doesn't forget what it is holding. (Normally, SRAM without a battery to help it, forgets everything when the power is turned off.) A tiny current is required to sustain the SRAM memory, usually on the order of 10s or 100s of microamps, and this level of current is almost "perfect" for a lithium button battery (which have shelf-lives in excess of 10 years.) The usual series of button batteries used for these purposes is a CR20xx type. They are all about the same diameter, but different thicknesses and therefore different "total energy" (how long they will last for you.) A CR2025 is thinner than a CR2032, for example, and holds a little less energy. I'm not sure what is used in NES cartridges, but I'd bet it is one of these types. They are pretty much a natural for this kind of use with SRAM.
The IBM PC also uses such batteries in order to "hold" the BIOS settings and the date and time, so that when you turn your computer back on it is able to restore the clock time correctly and boot up properly for you. Similar idea, different use.
If the save battery fails, it's not difficult to replace -- if you are able to open up the cartridge. You will pretty easily see the battery. Just replace it with a new one and that will repair the cartridge so that saved games stay saved, again.