I've had a few mentors ... DIR and non-DIR. What they had in common was patience, a desire to share what they knew, a positive attitude, patience, a sense of humor, and a lot of patience.
My first mentor was a fellow I was working with at the time. When I signed up for my class he was almost as excited as I was, and couldn't wait till I got my certification so he could teach me how to catch our local delicacy ... dungeness crabs. Did my first boat dive off his 12-foot aluminum fishing skiff. He was an old-school diver whose gear was almost as old as he was ... but a fun fellow to go share early diving adventures with.
My next mentor was a guy who I later took one of my earliest tech classes from. Met this fellow on a charter out to one of our premier dive sites (Possession Pt. Ferry). When he found out I was a new diver he asked me if I had a buddy yet. I said no, and he offered to "show me around". Took one look at my little C4 light and pulled this huge honkin' D8 out of his bag and said "Here, you'll need this". He was right ... of course. Looking back it still amazes me that the dude would have volunteered for what he had to know would be a very short dive with a newbie on a dive charter to one of the best sites in Puget Sound ... but that's what mentors do.
And then there was my DIR mentor ... a fellow who used to frequent ScubaBoard a lot ... Uncle Pug. He didn't just teach me the mechanics of good diving, he helped me learn how to THINK about diving. His guidance changed the way I dive, the way I think about diving, and the way I go about helping others learn to dive. That's what mentors do ... they help you learn things at a level that you simply can't get in a class.
Reading Lynne's post above, there's a continuity ... in that everything I helped her learn I had learned previously from the mentors I had. And I have watched her apply that same desire to "pay it forward" to others who have just started down the path to becoming better divers. And more recently I have noticed some of those she has helped begin helping others who are starting down that same path. That's what mentors do ... they "pay it forward".
Classes only teach you what you need to learn, and give you the tools to craft knowledge into skills. Mentors are the people who spend time in the water with you, guiding you through the process of applying that knowledge and developing the skills. Classes work in an artificial environment designed to achieve the objectives of the class. Mentors work in the real world, putting "ah HA" context around what you got from the class.
They are both important ... but they promote learning in very different ways.
Not every good diver can be a good mentor ... in fact, most cannot. Skills aren't what defines a mentor ... attitude is. You have to be someone who derives enjoyment from sharing, and takes pride in the accomplishments of the person you're helping, as well as their enthusiasm for things you may have experienced many time ... but are completely new to them.
Not everyone can do that. Not everyone wants to. Good mentors are like gardeners, deriving pleasure from the growth of others.
So to answer the OP's question, what to look for in a mentor ... look for someone who's going to truly enjoy the dive YOU want to do, not the one they think you should be doing. If your potential mentor is trying to talk you into doing a dive you don't feel ready for yet, that's not the right mentor for you. Look for someone who you will help you develop your skills at a pace you're comfortable with. Above all else, look for someone who wants to have FUN ... and who makes fun for both of you the priority. A mentor is, first and foremost, the person you should really ... REALLY ... look forward to doing your next dive with.
... Bob (Grateful Diver)