In addition to getting a tattoo that's nearly invisible by itself, you can also add to existing tattoos with blacklight ink. You can get a blacklight tracing or shading over a tattoo you already have so that when you get under a black light, it becomes a different tattoo.
Because the tracings of the tattoos are nearly invisible in daylight, there is potential for writing hidden messages that are only visible under ultraviolet lights. You could also pair these next to words you have tattooed in traditional ink. Hidden messages, phrases or symbols could be rebellious or funny or special only to a few people you know.
An interesting trend in blacklight tattoos is the practice of getting anatomical tattoos on the skin that reflect what's beneath it. For instance, you could get a full arm sleeve tattoo showing the muscles and tendons in your arm under a black light. Bones, veins, or organs would make for interesting scientific blacklight tattoos.