For some people, it's the particular culture or group of people featured in the author's novels. I was talking with a friend who says that's one reason for which he likes Daniel Silva's Gabriel Allon series. He likes the Israeli Jewish culture that's an important part of the series. I see his point. I've always liked Tony Hillerman's Jim Chee and Joe Leaphorn series in part because the Navajo culture (and some other Native cultures, too, such as the Hopi and Ute) are as integral to the books as the plot is. I've learned a lot from reading those series.
Other people come back for more because they have a particular liking for the sleuth; I've blogged about the sleuth before, so I won't go on about it here except to say that sleuths make an important difference in the appeal of a series. I like sleuths to be likeable, or at least to be compelling. That's another interesting aspect of the Gabriel Allon series; Allon is an assassin who's undercover as an art restorer. As my friend pointed out, that juxtaposition of roles Allon plays makes him a fascinating character.
For me, one other aspect of a compelling series - the kind that makes you come back for more - is that the series makes you think. It doesn't insult the intelligence. For instance, Patricia Cornwell's Kay Scarpetta series offers a lot of interesting information about forensics. In fact, Patricia Cornwell, as most fans know, worked for the Office of the Chief Medical Examiner of the Commonwealth of the Virginia. Her expertise is clear in the Kay Scarpetta novels. The same is true of Robin Cook's novels. Cook's Laurie Montgomery/Jack Stapleton series is full of interesting medical knowledge. It doesn't insult the reader, either. Cook also gives the reader food for thought, since his novels deal with sometimes-complicated ethical issues. Michael Crichton's mysteries (A Case of Need, Airframe and Prey, for instance) also presented important "food for thought." They weren't in series form, but I think that the point about appealing to the mind still applies.
In fact, that's one reason for which my sleuth, Joel Williams, is an eductor. That's my area of expertise, so I have something to share with the audience. I think it's important for a mystery author to add something interesting to her or his stories. It makes for better writing (and, hopefully, reading) if the author has some knowledge to share along with the mystery.
There are, of course, lots of other things that bring a mystery reader back again and again. I thoroughly enjoy the English-village setting of Caroline Graham's Inspector Barnaby series. I've also come back again and again because Barnaby himself is a likeable sleuth. Colin Dexter's Inspector Morse series has always appealed to me, too, because of its Oxford setting. Higher-education settings interest me; I've spent a lot of years on campuses. That's, in part, why my own Joel Williams series takes place in a university setting.
I also like the fact that Inspecter Morse is not a one-dimensional character; he is complicated and sometimes self-destructive, but never boring. Multi-dimensional characters such as Morse or Gabriel Allon, even if they aren't the sleuth, can also bring a reader back for more.
In the end, different people return to a series for different kinds of reasons. What is your favorite series? What's brought you back for seconds?