The most recent results of searches at the LHC for the Higgs boson have turned up possible hints of such a particle with mass about 125 GeV consistent with standard model (SM) expectations. This has many potential implications for the SM and beyond. We consider some of them in the contexts of a simple Higgs-portal dark matter (DM) model, the SM plus a real gauge-singlet scalar field as the DM candidate, and a couple of its variations. In the simplest model with one Higgs doublet and three or four generations of fermions, for mass the invisible decay tends to have a substantial branching ratio. If future LHC data confirm the preliminary Higgs indications, will have to exceed . To keep the DM lighter than , one will need to extend the model and also satisfy constraints from DM direct searches. The latter can be accommodated if the model provides sizable isospin violation in the DM-nucleon interactions. We explore this in a two-Higgs-doublet model combined with the scalar field . This model can offer a 125 GeV SM-like Higgs and a light DM candidate having isospin-violating interactions with nucleons at roughly the required level, albeit with some degree of fine-tuning.