I have a scene all drawn up that involves some non-traditional temporary tattoos for our crew, I just don't have a story to fit it into yet and it's definitely begging for one.
As for a traditional western tattoo? To me it would depend on what the meaning and cultural significance of those kinds of tattoos would be in the 22nd century. If it was the same as today ... hm.
I can't help thinking about all the four Human guys of the NX01 gang:
I can see Malcolm getting one, easily. There's just something about it that fits the time he finally made his decision to go with Starfleet instead of the Royal Navy. Or later, when he joined the Section. It's got that secretive, symbolic, slightly rebellious nature to it that I think fits.
On the other hand, I don't see Archer getting one. Something about his upbringing and experience seems completely the opposite - like as if he always had a lot more control and never rebelled in a conscious way. He just broke the rules and got away with it because he was a natural-born leader. I don't think he'd be as interested in a tattoo, or that it would mean as much to him. He might do it if it was important enough to something diplomatic with far-reaching implications.
Travis? Yes, actually, if it had something to do with Boomer culture or was something historical and related to fly-boy, pilot culture. It would be a symbol of belonging to a group or kin of some kind. Identity-related and it would have a lot of meaning to him.
Which brings me back to Trip. I think that the "accidental" (i.e. drunk or something) scenario is good.
But deliberately? I think it would be - like Kathy said - for emotional reasons and as a way of remembering something very important to him. Or it might be as part of a ritual (Vulcan for example) that was important to someone else - again, for symbolic or emotional reasons. Trip was often the emotional thermometer for for the gang, so it seems appropriate.