To do one’s level best is to do the best that one can do. The expression is of American origin and dates from the mid-19th century. Why level? There are two theories about this. The first is that the reference is to be level, straight or truthful and the second is that it derives from panning for gold in the Californian gold rush, where a level pan gave the best chance of spotting gold. The latter theory may indeed be the correct one because to level with someone i.e. to tell the truth, is indeed American but only dates from the 1930s, whereas the panning for gold theory fits the mid-19th century origin for doing one’s level best.