Galactic field black hole (BH) low-mass X-ray binaries (LMXBs) are believed to form in situ via the evolution of isolated binaries. In the standard formation channel, these systems survived a common envelope phase, after which the remaining helium core of the primary star and the subsequently formed BH are not expected to be highly spinning. However, the measured spins of BHs in LMXBs cover the whole range of spin parameters. We propose here that the BH spin in LMXBs is acquired through accretion onto the BH after its formation. In order to test this hypothesis, we calculated extensive grids of detailed binary mass-transfer sequences. For each sequence, we examined whether, at any point in time, the calculated binary properties are in agreement with their observationally inferred counterparts of 16 Galactic LMXBs. The "successful" sequences give estimates of the mass that the BH has accreted since the onset of Roche-Lobe overflow. We find that in all Galactic LMXBs with measured BH spin, the origin of the spin can be accounted for by the accreted matter, and we make predictions about the maximum BH spin in LMXBs where no measurement is yet available. Furthermore, we derive limits on the maximum spin that any BH can have depending on current properties of the binary it resides in. Finally we discuss the implication that our findings have on the BH birth-mass distribution, which is shifted by ~1.5M_{sun}_ toward lower masses, compared to the currently observed one.