Old English wegan "find the weight of, have weight, lift, carry," from Proto-Germanic *weganan (cf. Old Saxon wegan, Old Frisian wega, Dutch wegen "to weigh," Old Norse vega, Old High German wegan "to move, carry, weigh," German wiegen "to weigh"), from PIE *wegh- "to move" (cf. Sanskrit vahati "carries, conveys," vahitram "vessel, ship;" Avestan vazaiti "he leads, draws;" Greek okhos "carriage;" Latin vehere "to carry, convey;" Old Church Slavonic vesti "to carry, convey;" Lithuanian vezu "to carry, convey;" Old Irish fecht "campaign, journey").
The original sense was of motion, which led to that of lifting, then to that of "measure the weight of." The older sense of "lift, carry" survives in the nautical phrase weigh anchor. Figurative sense of "to consider, ponder" (in reference to words, etc.) is recorded from mid-14c.