Well Chelsey, I'll tell you what I told my daughter at the same age when this question came up. First, consider a tattoo as a permanent change in your appearance. You may feel differently about it someday, so if you decide to get one, be sure it's some place that you can easily hide under your clothes. Second, tattoos sometimes degrade over time. A lovely rose on your shoulder may turn into a shapeless blob in a few decades, another reason to position it so it can be easily hidden. Third, tattooing hurts. Fourth, there are no legal regulations in place in most jurisdictions about things like standards of hygiene and care for tattoo artists, so there's always at least a small risk of picking up some nasty infection.
None of those are reasons not to get a tattoo, they're just reasons to choose carefully what to get engraved on your body, where to put it, and who's to do it. Since you're wanting to honour your grandmother with this, though, I'd suggest you ask her what she thinks of it. There's probably something she'd like better.