I have a 9n Ford that was 6 volt and I put a 12 volt battery in it. I changed the starter relay to 12V (no solenoid). I changed out the generator for a 12V alternator (bought a kit including alternator, proper pulley, bracket, etc.) At first, I added a resistor in the ignition but it needed a new set of points every year. So finally I updated it to a 12v coil and electronic ignition. It has been working this way for several years now.
That's a lot of work and fair bit of change, so to answer your original question, yes you could put a small 12v solar panel on it. They typically only put out a fraction of an amp, and will work just fine at 6V. I don't have a good answer as to how long your battery will last, my experience with batteries has been all over the map. But it works for your friend, it should work for you as well.
There are also some simple charge regulator circuits that could be built with an LM317 and a few components. I think there's one in the LM317 datasheet, I don't know if you're into building your own or not. It could be powered from its own solar panel or powered off of your existing 12v system.