Results are reported from a search for non-standard-model Higgs boson decays to pairs of new light bosons, each of which decays into the oppositely charged dimuon final state. The new bosons may be produced either promptly or via a decay chain. The data set corresponds to an integrated luminosity of 5.3 inverse femtobarns of proton-proton collisions at sqrt(s) = 7 TeV, recorded by the CMS experiment at the LHC. Such Higgs boson decays are predicted in several scenarios of new physics, including supersymmetric models with extended Higgs sectors or hidden valleys. Thus, the results of the search are relevant for establishing whether the new particle observed in Higgs boson searches at the LHC has the properties expected for a standard model Higgs boson. No excess of events is observed with respect to the yields expected from standard model processes. A model-independent upper limit of 0.78 +/- 0.05 fb on the product of the cross section times branching fraction times acceptance is obtained. The results are applicable to a broad spectrum of models and are also compared with the predictions of two benchmark scenarios as functions of a Higgs boson mass within the range 86-150 GeV and of a new light boson mass within the range 0.25-3.55 GeV.