We search for B meson decays into two-body combinations of eta, eta', omega, and phi mesons from 89 x 10(6) BB pairs collected with the BABAR detector at the PEP-II asymmetric-energy e+e- collider at SLAC. We find the branching fraction B(B0-->etaomega)=(4.0(+1.3)(-1.2)+/-0.4)x10(-6) with a significance of 4.3 sigma. For the other decay modes we set the following 90% confidence level upper limits on the branching fractions, in units of 10(-6): B(B0-->etaeta)<2.8, B(B0-->etaeta')<4.6, B(B0-->eta'eta')<10, B(B0-->eta'omega)<2.8, B(B0-->etaphi)<1.0, B(B0-->eta'phi)<4.5, and B(B0-->phiphi)<1.5.