In my research area one seminal reference is H. Bethe, ''Zur Theorie der Metalle'', Z. Phys. 71 205 (1931), see also the English translation by T. C. Dorlas (2009). On page 206 of the original reference we read
Solange von der Wechselwirkung abgesehen wird, gibt es für jedes Atom zwei Zustände mit gleicher Energie: Der Spin des Leuchtelektrons kann nach rechts oder nach links zeigen.
Thus it seems that at the time, at least in the German-speaking scientific community, the terminology spin 'left' and 'right', rather than 'up' and 'down, was used for the two basis vectors of the spin-1/2 irrep of $su(2)$. Was this indeed common practice? If so, was this the original terminology used? And why and by whom was it changed to 'up' and 'down'?
NB. I can speculate that the following happened. 'Left' and 'right' may originate from the Stern--Gerlach experiment, of course depending on the direction of the external magnetic field in the original experiment. Later on, '$\leftarrow$' and '$\rightarrow$' might then have been replaced by the more compact '$\uparrow$' and '$\downarrow$' for typographical reasons.