Different things for different materials probably. I know that in the wreck of the boat 'Ulu Burun' dated to the Bronze Age (around 1500 BC in the Mediterranean, I believe) there were what have been interpreted as weights.

Weights probably developed earliest in international trade and exchnage and only later developed for intra-regional exchange - i.e. trade of butter between villages or something - when
a) Market systems fully replaced barter economies, which may be quite recent
b) Effective refrigeration methods were made to make trading perishable things (like butter) over distances possible.

A weight is a universal measurement that works between social/cultural groups. Measurements like a 'yard' are specific to groups and therefore would not have been practical to abandon until economics - trade- necessitated it.

That's what I think, anyway.