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Pseudo-Nambu-Goldstone bosons (pNGBs) are naturally light, spin-zero particles that can be interesting dark-matter (DM) candidates. We study the phenomenology of a pNGB θ associated with an approximate symmetry of the neutrino seesaw sector. A small coupling of θ to the Higgs boson is induced radiatively by the neutrino Yukawa couplings. By virtue of this Higgs-portal interaction, (i) the pNGB acquires a mass m_{θ} proportional to the electroweak scale, and (ii) the observed DM relic density can be generated by the freeze-in of θ particles with mass m_{θ}≃3  MeV. Alternatively, the coupling of θ to heavy sterile neutrinos can account for the DM relic density, in the window 1 keV ≲m_{θ}≲3  MeV. The decays of θ into light fermions are suppressed by the seesaw scale, making such pNGBs sufficiently stable to play the role of DM.