We present a theoretical description of the thermopower due to magnon-assisted tunneling in a mesoscopic tunnel junction between two ferromagnetic metals. The thermopower is generated in the course of thermal equilibration between two baths of magnons, mediated by electrons. For a junction between two ferromagnets with antiparallel polarizations, the ability of magnon-assisted tunneling to create thermopower S-AP depends on the difference between the size Pi(up arrow,down arrow) of the majority- and minority-band Fermi surfaces and it is proportional to a temperature-dependent factor (k(B)T/omega(D))(3/2) where omega(D) is the magnon Debye energy. The latter factor reflects the fractional change in the net magnetization of the reservoirs due to thermal magnons at temperature T (Bloch's T-3/2 law). In contrast, the contribution of magnon-assisted tunneling to the thermopower S-P of a junction with parallel polarizations is negligible. As the relative polarizations of ferromagnetic layers can be manipulated by an external magnetic field, a large difference DeltaS=S-AP-S(P)approximate toS(AP)similar to-(k(B)/e)f(Pi(up arrow),Pi(down arrow))(k(B)T/omega(D))(3/2) results in a magnetothermopower effect. This magnetothermopower effect becomes giant in the extreme case of a junction between two half-metallic ferromagnets, DeltaSsimilar to-k(B)/e.