Where does the lithium in Salar de Uyuni comes from?
Due to its increasing economic importance, Lithium deposits are coming under more intense scrutiny, by organisations such as the United States Geological Survey (1).  The Salar de Uyuni deposit in Bolivia is one of the deposits of lithium brine that is undergoing modelling and research.

The USGS noted that all lithium brine deposits have some very similar characteristics, climatically, they are in arid environments, be near one or more aquifers and have had sufficient time for the accumulation of the lithium from weathering processes, other commonalities include the tectonic settings, associated volcanic/hydrothermal activity and local petrology (lithium bearing rocks) (1).

The geological setting of the Salar de Uyuni is described by the USGS (1) as being situated in the Altiplano of Bolivia and Chile - a high, internally drained plateau along the crest of the Andean convergent boundary orogen, that includes closed basins. Active faulting is present, causing in subsidence, allowing sediments to accumulate.

Observations made by Hofstra et al. (2013) (2) in similar lithium brine deposits in the United States came to the conclusion that melt inclusions in quartz phenocrysts in rhyolite tuffs in association with hydrothermal deposits of tin, molybdenum and beryllium have been found to be heavily enriched in lithium (to the order of 1000's ppm).  Hofstra et al. (2013) surmise from these observations that the source of the lithium has been leached out over time from the nearby rhyolitic tuffs of the Bolivian tin belts to the east of the Salar de Uyuni (2),(3). 

The lithium is then leached out of the rhyolite by meteoric water, draining into the basins and concentrated by evaporation in the local arid environments.

References

(1) Bradley et al. 2013, A Preliminary Deposit Model for Lithium Brines, USGS Open-File Report 2013–1006

(2) Hofstra et al. 2013, Silicate Melt Inclusion Evidence for Extreme Pre-eruptive Enrichment and Post-eruptive Depletion of Lithium in Silicic Volcanic Rocks of the Western United States: Implications for the Origin of Lithium-Rich Brines*, Economic Geology

(3) Arce-Burgoa and Goldfarb, 2009, Metallogeny of Bolivia, Metalliferous Ore Deposits of Bolivia