I'm going to leave the jokes about doing a fractal tattoo the same way a fractal is plotted (iteration by iteration) aside.
What you're going to want to do is find a tattoo artist that does a lot of very fine detail work, ideally using shading and membrane needles. Shading is probably the best way to depict a fractal image on skin, because, by definition, doing it point by point just isn't going to cut it. Ultimately it's an approximation on a rather low res medium (skin).
You're going to want to pick a fractal that will print well onto tattoo transfer paper. I don't know if freehanding it is going to be a good idea, or even feasible. My guess is that Sierpinski fractals would be your best bet.
If you're set on a Mandelbrot, maybe look for an artist that does a lot of oceanic depictions. It seems like if you're familiar with tattooing tentacles a lot, tattooing those parts of a Mandelbrot might not be too different.
What I would do is have the tattoo "signed" with the equation and parameters used to generate the fractal in question. But that's just me, do what you like.