Finger tattoos are small by nature, so simple symbols work best: a moon, a star, an initial, a short word across the knuckles, or a band that wraps the finger like a ring. Put designs on the top of the finger, between the knuckles, rather than on the sides or the palm side, where the skin holds ink much less reliably.
Fingers are the hardest place on the body to keep a tattoo crisp. The skin renews quickly and is washed, flexed and rubbed all day, so ink spreads and fades, sometimes within the first year. Some lines fall out during healing. Many artists will tell you up front that they cannot guarantee a finger tattoo, and a touch-up is normal rather than a sign of bad work.
Finger tattoos are always on show, and only gloves hide them. Many artists won't put a first tattoo on the fingers or hands and treat them as job stoppers. A tiny piece near the base of a finger, where a ring would sit, is the easiest to hide.
Healing is quick for such small pieces, but you will be washing your hands constantly. Pat the tattoo dry, moisturize thinly, and keep rings, gloves and soaking water off it for the first couple of weeks. Check it once the surface has healed, because that is when any patchy lines show up.