An interesting subject .. why Dram? when Dram is technically such a little measurement ... I have a theory on this and that is all it is as I have nothing to base it on only the charachter of a people. That charachter would be of both the Scotch and Irish people which in ways is so closely linked but with obvious differences.
Both cultures are generally known for their warmth, welcoming and joviality. But there is also another charachter which is not as obvious. It is an understated politness amongst themselves which was very much present in the bygone days.
In Ireland we use the word drop and probably can be directly swapped for Dram in Scotland, but it is not used as much nowdays like Dram is. But my grandfathers era whould always use drop especially in regards to a "drop of the craythur"
Example would be if people were visiting for whatever reason it was always seen as polite to offer a drink to you visitor ... however it was also polite to retort "only a small drop" ... so in Scotland I would imagine it would end up a person saying something "only a wee dram". However the politness would be returned once again by pouring a drop which ended up being a large one. Hence the dram was always much more than the actualy physical measurement.
However there maybe a more sinister reason for all this politness. If there was any occasion being held by a family and the neighbours were invited there was always the presure of putting the best foot forward, especially with funerals (wakes) & weddings. Therefore it was expected of you to offer a visitor a drink and this became the norm. However the visitor could not be seen to be there for the free drink and would reciprocate in the politest of fashions by asking for a small drink and not look like a greedy so and so infront of all the other persons present. But the final presure was how generous one's drop would be ... nothing like small town chat from your neighbours to put on the pressure.