The Higgs boson may decay mainly to an invisible mode characterized by missing energy, instead of the standard model channels. This is a generic feature of many models where neutrino masses arise from the spontaneous breaking of ungauged lepton number at relatively low scales, such as spontaneously broken R-parity models. Taking these models as framework, we reanalyze this striking suggestion in view of the recent data on neutrino oscillations that indicate nonzero neutrino masses. We show that, despite the smallness of neutrino masses, the Higgs boson can decay mainly to the invisible Goldstone boson associated to the spontaneous breaking of lepton number. This requires a gauge singlet superfield coupling to the electroweak doublet Higgses, as in the next to minimal supersymmetric standard model scenario for solving the mu problem. The search for invisibly decaying Higgs bosons should be taken into account in the planning of future accelerators, such as the Large Hadron Collider and the Next Linear Collider.