We present new Keck I/HIRES observations at R = 45,000 (=3 pixels) of seven stars near the turnoff of the old, metal-poor globular cluster M92. In three of these stars, we have signal-to-noise ratios (S/Ns) of 40 pixel-1, and in the other four, the S/N is near 20. The Li abundance in star 18 is high compared with the halo field-star plateau and is similar to that in the remarkable Li-rich halo field star BD +23°3912. In addition to the high Li abundance in star 18, there is a dispersion in Li abundance in our seven stars covering the full range of a factor of 3. We have attempted to determine whether the excess Li in star 18 is due to less than average Li depletion in this star from an even higher initial abundance, as predicted by the Yale rotational models, or whether it is due to the extraordinary action of Li production mechanisms in the material that formed this star. We have found no convincing evidence that favors Li production: (1) Stars 18, 21, and 46 have identical Ba abundances, which argues against Li production carrying an s-process signature. (2) These three stars have indistinguishable Ca, Cr, Fe, and Ti, which argues against supernova Li production. (3) We discuss ν-process production of Li and find no convincing observational evidence for this from the strengths of the Mg, Ca, and Fe lines. (4) The similarity in age of these cluster stars argues against cosmic-ray Li production that requires age differences of gigayears. The most likely explanation for the Li dispersion is differential Li depletion from a (possibly significantly) higher primordial Li abundance due to differences in the initial angular momentum in each star followed by spin-down; the most rapid rotators destroy the most Li, whereas the initially slower rotators preserve more Li.