What are we to make of Detective Loki?

I'm not sure if the following is sufficient as a complete answer already, but it should give some insights into how Det. Loki was intended and what he might have stood for, even if not giving too deep an analysis of his character and its meaning:

During his interview with Jake Hamilton Jake Gyllenhaal talks a bit about Det. Loki's character and what he meant to him:

For this movie there was the foundation of the script and it was a blueprint in a lot of ways. And when you work with a director like Denis Villeneuve, who directed this movie, he allows for that mystery to come out, so every character has been sort of developed separately with him. So when we come together in a scene, there are all these crazy masteries. And I think one of the great things about playing a detective and being the detective in this movie was that I created a lot of mystery for myself and in the character, so regardless of knowing what happens in the end as an actor or even as an audience member or objectively, I was always looking, I believed, in no matter what scene I was, that anybody could be the person who did this. And I knew that if I believed that, I knew the audience would believe it, because in a lot of ways Det. Loki is the eyes for the audience.

The biggest thing for me as an actor walking into every scene was, that I knew that Det. Loki would be listening a lot. And I knew that because of that I wanted to give him something to sort of rub up against. And if his directive is to always find that answer I wanted to have something that I was hiding. And so for me there was a past for him, a past of, I think, he went through the juvenile detention system for a while and I think he found himself in the institution of either maybe the military, but I doubt it in this case, more police work, and he found a family there, one that he hadn't had. And that past that he had, maybe he'd caused some trouble in his past, that past he's trying to cover up and in a lot of ways sort of tighten himself up and in the same time I think he's fascinated with the person who did this crime, the suspect, and their mind because he understands it. So I think he's looking for institutions, our director would say that all the time, that Loki represents an institution and I think that Hugh Jackman's character represents that sort of beating heart, that individual ideal where you're trying to get what you need as an individual and Loki represents that institution. And to me I decided that institution was not just, you know, clean-cut detective that in my opinion has become cliché.

There is also this review, which also presents some views on Det. Loki's character:

In Prisoners, a movie replete with complicated, showy characters, Detective Loki—played by Gyllenhaal—is an indecipherable question mark. Solemn and, for the most part, non-emotive, he's harboring more than a few inner demons. It's obvious from the way he blinks incessantly, a nervous habit that could mean he's a former drug addict, or, at one point in time, suffered from some kind of psychosis. There's no question that he's a lonely guy. The first time we see Loki, he's eating alone inside an empty Chinese restaurant on Thanksgiving night, trying to scam extra fortune cookies out of the waitress.

The script, written by Aaron Guzikowski, doesn't provide any concrete background information about Loki. The rest of the bleak and engrossing film's tormented characters, on the other hand, are given more obvious layers, namely grieving father Keller Dover (Hugh Jackman), a blue-collar repairman whose young daughter and her neighborhood friend have been abducted. Loki's the acting detective on the case, and his no-fail record precedes him, even if Keller doesn't trust Loki's abilities and decides to take matters in his own dirty hands—e.g., kidnapping the lead suspect, the brain-damaged Alex Jones (Paul Dano), boarding him up inside an abandoned building's decrepit bathroom, and torturing him.

There are moments early into Prisoners where you're left frustrated, wishing that Loki would show some kind of emotion. After all, he's investigating the possible homicides of two girls who haven't even entered grade school yet. Yet Gyllenhaal doesn't let you hate Loki, or even scoff at his monotone approach to crime-solving. There's an underlying melancholy to the character visible in Gyllenhaal's eyes —he's on the verge of breaking down, it's only a question of when. And then the catalyst for self-implosion happens in one particularly brutal scene, and that's it. Loki snaps, and it's animalistic and sorrowful. He's hurting badly. Whether something's conjured up past memories or he's simply reacting in the moment doesn't matter. As Loki treats his desktop keyboard like a head-banging rocker would his guitar after an especially raucous performance, Gyllenhaal's tapping into something primal, something explosive. From that point on, he's the most compelling thing about Prisoners.

I think Loki primarily served as the voice of reason amongst all those other people in the story that were mostly driven by their emotions. And thus his Masonic ring for example could also have emphasized this aspect and his nature as a seeker of truth. In this way his name could indeed refer to the Norse god, when concentrating more on his intelligence and his out-of-the-box-thinking rather than his role as a trickster. Or this rather "heathen" name just helps to further distance him from all the other heavily religious characters.

It is also emphasized right from the start that he doesn't seem to have much of an actual family himself, seeing that he takes his Thanksgiving meal in an empty diner. In a story that relies on family ties and commitments to a large degree, this casts him even more as an outsider and thus as an objective observer, which in turn helps him steer clear of all the emotions involved in the case.

The tattoos, as Jake Gyllenhaal explains, emphasize his troubled past (since he "spent six years in the Huntington Boy's Home") and give him a bit of an unusual background compared to your average police detective. Yet I'm not sure if the particular tattoos have any further meaning. But Wikipedia says he has many religious tattoos, which would tie in with the many religious symbols used in the movie, as also stated in your other question. But what this says about Det. Loki I'm not sure, as I would have seen him as rather the only one not influenced by some kind of blind faith or fanaticism or otherwise unreasonable external control. But maybe he once was a more religious man in the past.