Rubbing alcohol can be found at a dollar store, so it is readily available. The reason they use alcohol is because it has certain bonding properties (the forces that hold molecules together) that other substances lack.
Nail polish remover is mostly acetone, which can release a lot of fumes in a room if used in large amounts and can cause headaches: not a good choice for a child's project.
Hydrogen peroxide doesn't give off fumes, but it will start to change into a new substance (water and oxygen gas) when left in the light, so it is not going to be consistent.
Plus, neither of these compounds has the same properties as alcohol, so they would not work as well. The alcohol will expand and contract with temperature changes, which is why it is a fine choice.
Actually, thermometers used to be made of mercury, but it is toxic and hard to dispose of, so even the commercial ones (like in schools) are alcohol based. They add some red dye to make reading the thermometer easier.