High-sensitivity measurements of the linearly polarized solar limb spectrum produced by scattering processes in quiet regions of the solar atmosphere showed that the Q/I profile of the lithium doublet at 6708 Å has an amplitude ~10-4 and a curious three-peak structure, qualitatively similar to that found and confirmed by many observers in the Na I D2 line. Given that a precise measurement of the scattering polarization profile of the lithium doublet lies at the limit of the present observational possibilities, it is worthwhile to clarify the physical origin of the observed polarization, its diagnostic potential, and what kind of Q/I shapes can be expected from theory. To this end, we have applied the quantum theory of atomic level polarization taking into account the hyperfine structure of the two stable isotopes of lithium, as well as the Hanle effect of a microturbulent magnetic field of arbitrary strength. We find that quantum interferences between the sublevels pertaining to the upper levels of the D2 and D1 line transitions of lithium do not cause any observable effect on the emergent Q/I profile. Our theoretical calculations show that only two Q/I peaks can be expected, with the strongest one caused by the D2 line of 7Li I and the weakest one due to the D2 line of 6Li I. Interestingly, we find that these two peaks in the theoretical Q/I profile stand out clearly only when the kinetic temperature of the thin atmospheric region that produces the emergent spectral line radiation is lower than 4000 K. The fact that such a thin atmospheric region is located around a height of 200 km in standard semi-empirical models, where the kinetic temperature is about 5000 K, leads us to suggest that the most likely Q/I profile produced by the Sun in the lithium doublet should be slightly asymmetric and dominated by the 7Li I peak.