We present updated measurements of the branching fractions for B0 meson decays to ηK0, ηη, ηφ, ηω, η′K0, η′η′, η′φ, and η′ω, and branching fractions and CP-violating charge asymmetries for B+ decays to ηπ+, ηK+, η′π+, and η′K+. The data represent the full data set of 467×106 BB̄ pairs collected with the BABAR detector at the PEP-II asymmetric-energy e+e- collider at the SLAC National Accelerator Laboratory. Besides large signals for the four charged B decay modes and for B0→η′K0, we find evidence for three B0 decay modes at greater than 3.0σ significance. We find B(B0→ηK0)=(1.15-0.38+0. 43±0.09)×10-6, B(B0→ηω)=(0.94-0.30+0.35±0.09) ×10-6, and B(B0→η′ω)=(1.01-0.38+0.46±0.09) ×10-6, where the first (second) uncertainty is statistical (systematic). For the B+→ηK+ decay mode, we measure the charge asymmetry Ach(B+→ηK+)=-0.36±0.11±0.03. © 2009 The American Physical Society.