Recently, the ATLAS and CMS collaborations reported exciting hints of a
Standard Model-like Higgs boson with a mass around 125GeV. A Higgs boson this
heavy is difficult to realize in conventional models of gauge mediation. Here
we revisit the lightest Higgs boson mass in "more generic gauge mediation,"
where the Higgs doublets mix with the messenger doublets. We show that a Higgs
boson mass around 125GeV can be realized in more generic gauge mediation
models, even for a relatively light gluino mass ~1TeV. We also show that the
muon anomalous magnetic moment can be within 1sigma of the experimental value
for these models, even when the Higgs boson is relatively heavy. We also
discuss the LHC constraints and the prospects of discovery.Comment: 28 pages, 7 figures. Corrections and references are adde