I recall seeing a hand-drawn picture of spec of a ring (maybe of $\mathbb Z$?) that had been passed around in the early days of the Zariski topology. Does anyone know where I can find a copy?
I think you mean this image by Mumford.
You are thinking of the picture of Spec($Z[X]$) in Mumford's Red Book of Varieties and Schemes. A reproduction is downloadable at the link posted in the other answer, and other (less famous) pictures of prime spectra appear in the Eisenbud and Harris introductory book on schemes.
Be sure not to miss this very interesting paper (in English) on "Mumford's treasure map" and related concepts: Lieven Le Bruyn: Un dessins d'enfants