Most of my mentor relationships have developed organically, and they tend to consist of one part friendship and one part coaching. A good mentor-mentee cocktail has several ingredients:
1) Chemistry - if you have no interest in talking to your mentor/mentee for an hour's time, a mentoring relationship will never work.
2) Trust - mentees share with mentors their struggles and the career development decisions they are pondering. In return, a mentor may share personal experiences to shed light on lessons learned or factors to consider. If either person has doubts about this information remaining confidential, the relationship will be short-lived.
3) Common Interests - if you have similar passions outside of work, you will have points of entry for conversations and ultimately more to talk about.
4) Reciprocation - my mentors ask me for just as much as I ask them. This is the cost of doing business. If I can't commit to something, I'm honest. On the other hand, I want them to be comfortable asking me for favors, and I will usually bend over backwards to please them.
One particularly fertile source of mentors for me has been the pool of would-be bosses. These are people who at some point I might have worked for. We hit it off, but for whatever reason I didn't end up working for them, and we've been in touch ever since. In most cases, I'm grateful that I never worked for them, because I get to have their friendship and guidance without the anxiety of knowing that they are rating my professional performance.
When my family and I were getting ready to return home from our seven-month stint living in India, I interviewed with one of my would-be bosses. She was on maternity leave but was willing to meet with me over video conference. After our meeting, I cried. We had such a candid and intimate conversation. I knew I could learn a ton from her, but I also knew that the job wasn't a fit. Months later, I saw her in the gym locker room before work. We realized that we lived around the block from each other. A week later, we shared a glass of wine one night after work while her husband was travelling.
Today, she's a close friend and remains a mentor. We admitted to each other that we were glad I never worked for her - it's one of the reasons our friendship has blossomed and her advice has been as candid and valuable as it has. Now she comfortably approaches me for help with certain situations too, and I am happy to oblige.