What equipment do I need to reprogram an old garage opener remote?
Probably that is not possible.


There will not be a USB connection.
Even if you can find the microcontroller, you need to hard-solder the pins to be able to reprogram it.
If you are able to find a way to program it, you have to reverse-engineer the circuit to see what you can do.
Not all components might be identifiable (because they are hidden or even texts removed).


However, if you can find a way to access the pins to program, to solder wires to it, to see how the circuit is laid out, than it should be possible. Also note, that you are stuck to the hardware already present (unless you can add hardware yourself).

Update:

See also comments below (so check if you really have a microcontroller or a dedicated flashed all-in-one-chip).

Another problem is that for e.g. Arduino you can find easily a workable IDE. For a not so known brand, it might be hard to set up the IDE, find documentation, user support etc.