We design functionalized nanopores in graphene monolayers and show by molecular dynamics simulations that they provide highly selective passage of hydrated ions. Only ions that can be partly stripped of their hydration shells can pass through these ultrasmall pores with diameters of similar to 5 angstrom. For example, a fluorine-nitrogen-terminated pore allows the passage of Li+, Na+, and K+ cations with the ratio 9:14:33, but it blocks the passage of anions. The hydrogen-terminated pore allows the passage of F-, Cl-, and Br-anions with the ratio 0: 17:33, but it blocks the passage of cations. These nanopores could have potential applications in molecular separation, desalination, and energy storage systems.