Helium core white dwarfs (WDs) with mass M <~ 0.20 M sun undergo several Gyr of stable hydrogen burning as they evolve. We show that in a certain range of WD and hydrogen envelope masses, these WDs may exhibit g-mode pulsations similar to their passively cooling, more massive carbon/oxygen core counterparts, the ZZ Cetis. Our models with stably burning hydrogen envelopes on helium cores yield g-mode periods and period spacings longer than the canonical ZZ Cetis by nearly a factor of 2. We show that core composition and structure can be probed using seismology since the g-mode eigenfunctions predominantly reside in the helium core. Though we have not carried out a fully nonadiabatic stability analysis, the scaling of the thermal time in the convective zone with surface gravity highlights several low-mass helium WDs that should be observed in search of pulsations: NLTT 11748, SDSS J0822+2753, and the companion to PSR J1012+5307. Seismological studies of these He core WDs may prove especially fruitful, as their luminosity is related (via stable hydrogen burning) to the hydrogen envelope mass, which eliminates one model parameter.