I, also, would advise against a name, even your child's.
A month ago I had tattoo no. 3 done. A bracelet on my right wrist with the astrological symbols for DH and I. He almost died 2 years ago, and I've been thinking about it ever since. But even though I am besotted with my DH and even though I had to have 6 months of therapy after he came out of hospital as I was so traumatised by what happened to him ...
Even taking that into account, I STILL didn't want his name tattooed on me, but the astrology symbols still make it personal enough to me and him.
Last week, after also considering it for a long time, he had a tattoo on his upper right arm to look like the skin has been peeled back to expose the bone, and our wedding anniversary date is "carved" into the bone like a scrimshaw, and then our astrology symbols are like cuts below it.
Any tattoo artist worth their salt will expect you to make an appointment, and pay a deposit. These are both a good idea, as it ensures you are really committed.
Ask tattooed people you know if they'd recommend someone, or scout out different artists to see whose work you like. If you live in the East Mids, PM me and I'll tell you my guy.
Think about a symbol that represents your DC for you, even if it's a daisy or a Matchbox car.