It's actually an interesting unit: a ton of air conditioning is equal to the cooling that you get from melting a ton of ice per day. It has nothing to do with the weight of the air conditioner. Presumably this was useful in the olden days when businesses used ice for cooling and wanted to know what size of refrigeration unit they would need.
One ton of cooling is defined as 12,000 BTU/hour. The "real" value is 11966; apparently it's coincidental that it is so close to a round number. (One ton of refrigeration is 3517 watts in metric.)
No, the latent heat of fusion is 143.6 BTU/pound, not 144. It's just a coincidence that the energy to raise water one °F ÷ the energy to melt ice is almost 144.
Like many customary units, the ton (of cooling) is tied to historical accidents and I dare say is easier to imagine for a layperson; one could visualize the coolness rushing off a cubic-meter block of ice in a room, and how quickly it would melt depending on the temperature. 12000 BTUs, in the other hand, is not as intuitive (and the BTU itself is a customary unit, which is the amount of energy required to raise the temperature of one pound of water by one degree Fahrenheit).
And further, I guess there's just not much to be gained by going metric and saying "14 kilowatt AC unit" vs. "4-ton AC unit".
When picking an AC unit, running cost is not the primary concern: sizing is the primary concern. If you pick too large or small of a unit you will have issues. Too small won't be able to cool, too large generally won't be able to bring humidity down (outside of the newer inverter units that can run slower for longer).
Tons (or, BTUs of cooling) makes more sense, because the efficiency for a given cooling amount is variable.
For efficiency, we have efficiency ratings in the SEER/SEER2 ratings.
