We report the discovery of eight lithium-rich field giants found in a high-resolution spectroscopic sample of over 700 metal-poor stars ([Fe/H] < -0.5) selected from the Radial Velocity Experiment survey. The majority of the Li-rich giants in our sample are very metal-poor ([Fe/H] ≲ -1.9), and have a Li abundance (in the form of 7Li), A(Li) = log (n(Li)/n(H)) + 12, between 2.30 and 3.63, well above the typical upper red giant branch (RGB) limit, A(Li) < 0.5, while two stars, with A(Li)∼1.7-1.8, show similar lithium abundances to normal giants at the same gravity. We further included two metal-poor, Li-rich globular cluster giants in our sample, namely the previously discovered M3-IV101 and newly discovered (in this work) M68-A96. This comprises the largest sample of metal-poor Li-rich giants to date. We performed a detailed abundance analysis of all stars, finding that the majority of our sample stars have elemental abundances similar to that of Li-normal halo giants. Although the evolutionary phase of each Li-rich giant cannot be definitively determined, the Li-rich phase is likely connected to extra mixing at the RGB bump or early asymptotic giant branch that triggers cool bottom processing in which the bottom of the outer convective envelope is connected to the H-burning shell in the star. The surface of a star becomes Li-enhanced as 7Be (which burns to 7Li) is transported to the stellar surface via the Cameron-Fowler mechanism. We discuss and discriminate among several models for the extra mixing that can cause Li production, given the detailed abundances of the Li-rich giants in our sample.