Simon Tatham has prepared a Hasse diagram of the medal table. The main idea is:
So we want to say that one country has done strictly better than another if the medal score of the latter can be transformed into the former by a sequence of medal additions and medal upgrades.This gives a partial order on the countries.
Alternatively, we could say country A does strictly better than country B if and only if A gets more points than B under all weighting schemes in which we assign x points for a gold, y points for a silver, and z for a bronze, with x ≥ y ≥ z ≥ 0. This seems like it's equivalent to Tatham's order, but I haven't thought that hard about it.
One could extend this to include the population of a country; the natural order there would be that A did strictly better than B if the medal score of B can be transformed into that of A by a sequence of medal additions, medal upgrades, and population reductions. The idea here, of course, is that if two countries win the same assortment of medals, the one with lower population did better. But going there is dangerous; do we then take into account GDP? Popularity of sports in general in the country? The fact that the particular set of sports in the Olympics is more popular in some countries than others?