The second solar spectrum offers a novel diagnostic window not only to explore the magnetic field of our nearest star, but also to determine the atomic and isotopic abundances in the solar atmosphere. A very interesting example, concerning this second aspect, is the Li I resonance doublet at 6708~Å. Here we present a detailed theoretical investigation, carried out within the framework of the density matrix formalism, on the atomic physics that is involved in the generation of the observed Q/I profile, which is the result of a complex blend of the D_1 and D_2 lines of the two stable isotopes of lithium (^6Li and ^7Li), both showing hyperfine structure. We discuss the diagnostic potential of this linear polarization profile for the determination of the lithium isotopic abundance in the quiet Sun, and we investigate the magnetic sensitivity of this profile to a microturbulent magnetic field.