Firstly, there is nothing in the books -- canon -- to say he had tattoos. It's just a film thing, but it's one of those film things that I, personally, have never seen a major problem with using. It just seems like a Sirius kind of thing to do, really. He already does the rebellious thing, has the flying motorbike, etc.
Are you wanting the reasoning for tattoos to be nonchalance or humourous, or having a bit more of a seriousness to it? Rebellion and freedom could have been used after he ran away from home, for instance, and at an age of 16/17, there's still enough of that youthful rashness for Sirius (of course, not that he really seems to grow *out* of that rashness, either).
I don't really know anything specific, right now, as I've never given thoughts on what the tattoos would be or for why, exactly. It's just one of those things that, even though it's not canon, technically, I've never had a problem believing it if a writer uses it.