As Spaw (and others) note, the main advantage of nitrogen is that it's expected to by dry. Air, compressed into the tank, is nearly always at close to 100% relative humidity (at the conditions inside the tank). Some of the moisture can be dropped out when you expand it on the way to the tire, but it's always going to be "a bit damp" in the tire.
People commonly use nitrogen in aircraft tires largely to assure that the filler is dry, and in fact one can order "ordinary" nitrogen, or you can specify "dry nitrogen" as a separate product from commercial suppliers.
If moisture condenses inside a tire in flight, and freezes in a lump, touchdown and spinning the wheel up to 90 to 200 kt with a lump of ice in the bottom of the tire can be "bumpy," especially with large tires.
If you get any similar ice in an auto or other "ground vehicle" tire, it will happen while the vehicle is sitting, and will be broken up and distributed before you reach speed where it matters much. The "set" in the rubber will cause more out of balance than any expected ice - for any normal operation.
As to using hydrogen or helium, one would expect a slight reduction in inflated weight with a gas with lower molecular weight. Helium, however, is extremely difficult to contain over long periods except with container designs that specifically address its "peculiarities." Even ordinary glass is permeable to helium, albeit usually at slow leakage rates. Expecting helium to stay inside an ordinary automobile tire is about like believing in pixie dust.
Hydrogen actually is a little bit easier to keep in a can, but still would require much more frequent replenishment than a heavier gas.
The bottom line though is that if your tire can't stand exposure to the gas on the outside of the tire with satisfactory performance, a different gas on the inside isn't going to provide a lot of difference. Unless your special use demands that you squeeze out every little bit you can get, 'taint worth the hassle - or the expense.