We report a high-resolution neutron diffraction study on the orbitally-degenerate spin-1/2 hexagonal antiferromagnet AgNiO2. A structural transition to a tripled unit cell with expanded and contracted NiO6 octahedra indicates root(3) x root(3) charge order on the Ni triangular lattice. This suggests charge order as a possible mechanism of lifting the orbital degeneracy in the presence of charge fluctuations, as an alternative to Jahn-Teller distortions. A novel magnetic ground state is observed at base temperatures with the electron-rich S = 1 Ni sites arranged in alternating ferromagnetic rows on a triangular lattice, surrounded by a honeycomb network of non-magnetic and metallic Ni ions. We also report first-principles band-structure calculations that explain microscopically the origin of these phenomena.