Now I'm aware that this may be a shot in the dark; but does anyone know if there's a tire company that makes rolling resistance tires for 12-inch rims? I currently looking for tires for a Geo Metro. If not, then what would be the next best alternative?
Narrower does not mean lower rolling resistance. The difference is not large, though. Small differences from width are likely to be overshadowed by going from one model of tire to another, which will have a much bigger difference. Additionally, new tires with full tread have been shown to have much more rolling resistance than worn tires.
Simplified explanation about width: With a given load and pressure, you're going to have a certain size of contact patch. Changing the width of the tire changes the shape of the contact patch, but not the size. If you make it narrower, the contact patch has to get longer. A large part (most?) of rolling resistance comes from the sidewall deformation required to make the contact patch. A wider tire has a shorter, wider contact patch, and therefore less sidewall deformation.
Looking at it another way: A narrower tire has a lower load capacity at a given pressure. Again looking at sidewall deformation, this means that a narrower tire with the same pressure and load is going to have more sidewall deformation because it will be at a higher percentage of its available load capacity.
It's far more complicated and not completely understood by me, but I have more theory than you could ever want to read and a few observations in the tire width link in my sig.
I've been browsing a book I got on modifying cars for handling recently, and it appears that rim width can have quite an effect on the characteristics of tires. (Yeah, great!, another variable) Putting too wide tires on too narrow a rim, as might happen when you undersize the rims then have to use relatively wide tires to get the diameter right, will mean they bulge out a bit giving less contact patch, but not necessarily giving less RR because deformation energy is increased. Hence if you pick say a 13" rim that normally would have 165/70R13 on it, and you have to have 185/75R13 to get close to your stock 195/70R14 diameter, then not only might you have worse RR than the 14", you might have worse grip than the 165/70R13...
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Considering the difficulty of finding ANY 12 inch tires, I wouldn't expect it's worth the effort or cost to try to find anything in specific...take what you can get.
If you're talking about upgrading to a larger wheel, I think that's going to make your rolling resistance worse; it almost always does. If you could get the wheels for free and having larger wheels made cheaper tires available (doubtful), then it would be worth it.