Adding to Sabre Tooth's summary:
In addition to weathering of felsic volcanics as ST describes, another aspect that has received attention over the years has been direct input to the salar (or playa) from hot spring sources. The idea being that the lithium is coming in as magmatic, not just meteoric water. There is no question that hot springs are typically associated with lithium brines - but since the causal relationship is not yet conclusive, there could be an underlying confounding variable.
A paper published recently by Daisuke Araoka (2013), notably covering the same part of Nevada as Hofstra (2013), finds isotopic evidence for the hot spring source over the weathering of felsic volcanics. So the debate is not quite over. There are some unusually high Li levels in silver and gold vein mineralization adjacent to Clayton Valley. I'm not sure how that fits in - the Ag-Au mineralization is quite a bit older but it could still relate to the bigger story. I don't know the Uyuni area as well as Argentine salares, but I don't think the hot spring model has been pushed there too much although I do hair it mentioned in Argentina occasionally.
A good geochemical discussion on sources in the Andes is provided in Risacher (2003). His work also has application in developing exploration techniques. The amount of research (aka funding) that has gone into porphyry deposit genesis and exploration methods dwarfs the amount that has gone into lithium brines. And much of the work that has been done has been by a few companies that have had no motivation to publish their work in the way that the academic groups researching porphyries have. Consequently there is still work to be done in this area. Risacher's paper, and others by him on the same subject are readable and interesting examples of applied geochemistry in genetic models and potential exploration methods.
Araoka, D., Kawahata, H., Takagi, T., Watanabe, Y., Nishimura, K., Nishio, Y., 2013, Lithium and strontium isotopic systematics in playas in Nevada, USA: constraints on the origin of lithium, Mineralium Deposita, v, 49, is. 3, pp 371-379
Risacher, F., Alonso, H., Salazar, C., 2003, The origin of brines and salts in Chilean salars—A hydrochemical review: Earth-Science Reviews, v. 63, nos. 3–4, p. 249–293.