Peter Roquette has writteen a series of articles entitled
The Riemann hypothesis in characteristic p, its origin and development
several parts, total length close to 200 pages.
Bibliographic details and pdf are avalaible here: http://www.rzuser.uni-heidelberg.de/~ci3/manu.html (see papers 26,27,36).
I am not familiar with the content of these paper, the following is based on a very rough browsing.
The covered time span seems to be roughly 1920-40.
E. Artin's (1921) and F.K. Schmidt's (1925) theses seem the most important starting points. However, several other names are mentioned as well; H. Hasse worked on the Riemann Hyothesis (in this context) around 33/34; referring (freely translated) to Artin and Schmidt zeta-functions.
F.K. Schmidt: Zur Zahlentheorie in Koorpern der Charakteristik p. (Vorlaeufige Mitteilung.) Sitz.-Ber. phys. med. Soz. Erlangen 58/59 (1926/27) 159–172
is mentioned as the paper introducing the zeta-function for function fields (fin.gen. and trans. deg. 1).
As motivation for these investigations, the analogy with the number field case is mentioned.
ADDED: In retrospect, I am now unsure whether this is what was asked for. Sorry, in case it should not be.