Absolute *best* calibration for a thermometer is the "triple point" of water. You can make a triple point cell by boiling water in a test tube, then sealing the tube while the water is still boiling, so the water vapor displaces all the air. Now, when you cool the cell, you can witness water continuing to boil as it cools, because the pressure will drop in lockstep with the temperature -- and at 0.1 C (IIRC) you'll get to see boiling water and ice at the same temperature. That's the triple point, and it's solely a function of the compound being examined (assuming it's pure, of course). There are several semi-convenient triple points that are used to calibrate master thermometers; that of water, of course, is the most convenient, but the triple point of carbon dioxide is around -78 C, IIRC, that of sulfur dioxide is somewhat higher, and that of anhydrous ammonia a good bit colder.
You can also use the melting points of a couple metals as calibration temperatures -- mercury at -39 C and gallium at 21 C are the most convenient (though gallium is a little hard to get and has a very strong tendency to supercool). Melting points are harder to use with materials that have low latent heats of fusion (compared to water), but the problem with water is that the melting point is so strongly dependent on pressure -- pure water will melt at -15 C under relatively modest pressure (which is why ice skates work).
For the experiment you set up, however, the pressure is well controlled (there's much less than 0.1 C variation in melting point over the normal range of atmospheric pressure in the regions where 95% of humanity lives). The problem is that the supercooling will only give a dependable 0 C if the water doesn't all freeze when the supercooled state collapses (i.e. the water suddenly freezes). What I'd suggest is to wait for the thermometer to read about -3 C, then stir or introduce a small crystal of ice from the freezer as a nucleation point; you want to get a freeze to slush, not to a solid block, and thus get water and ice at the same temperature -- which, at ordinary ambient pressures, has to be very-very close to 0 C.
Using ice cubes in water might not work, because the water can be a lot warmer than the cores of the cubes, and water is such a poor conductor that you can have 5 C variation over a few mm.