Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people working with valuations (like Ostrowski, Krull, Dedekind, Hensel et al) would have started this idea. Just an idea.. there is a nice article on the history of valuation written by Peter Roquette, maybe I can find this there?

Edit: Maybe Keith is right. In fact I took a look at the text by Roquette and as far as I could see (I didnt read the whole text though) height was not mentioned.