Here I rely on Henry Alford's still valuable commentary. Up to his time (1850s), scholars had proposed at least eight different interpretations of this word in 2 Timothy. Its fundamental meaning is clear: "rightly cutting." But controversy arises as to how much of a metaphor Paul is making of it. The "rightly dividing" proposal comes from (a) sacrificial practice — knowing how to cut into the victim, how to separate out the relevant innards, etc., or (b) scriptural interpretation — knowing how much of a biblical verse to excerpt, at which points to begin and end your citation, etc.
You probably don't want my personal opinion, but I'll mention anyway that his advice is to someone who's to be a good worker, an ἐργάτης. If you think 2 Timothy is written by Paul, and if you remember that Paul was a tentmaker by trade, you may be inclined to think that he used this term frequently in the process of cutting the cloth that was the raw material for the structures he created. "Measure twice, cut once" must have been for Paul just as axiomatic as it is for carpenters. Cutting according to the correct measure would produce a tent, or a gospel mission, that one wouldn't be ashamed of.