The best mentors aren't those who are super successful and noteable, they're the ones who have been through what you're about to do 3-5 years ago.
Using that filter, find companies that you admire the are growing and have a great brand around them - then approach the founders/CEO and ask them if they have 15 minutes for a call to talk about your business challenges.
If you don't have a business, maybe you can use a tough decision you're facing as the basis for reaching out - the key is to have a purpose and let them know why you think they can help.
If you present yourself right, then they'll usually ask that you keep in touch and let them know if they were helpful. Do that, follow up in 3 weeks - maybe in 2-3 months circle back and ask for another quick call or coffee as you'll be near by in a couple days (if you're not, change your schedule and meet face to face!)
Straight out asking "Will you be my mentor" is never a good idea. That relationships should just form naturally and it's on you to continue nurturing it over the years.
I've had many mentors over the past 10 years and I've never told them or asked them to by my mentor. I just turn to them for big / hard life decisions and always circle back and thank them for their time - when possible I mention them in press interviews - so they know how much it's appreciated.
They're reward is to see you implement their feedback and grow as an individual. If you do that, they'll always be there to take your call.
The question is still open after 2 months, so decided to answer :)
First rule: do not try "to find relevant" mentors - you will put a lot of effort and the result is not guaranteed. From my own experience - the best advise I have received sometimes came from unknown mentors. Still remember the case when I was participating in StartupWeekend and organizers asked: "Which mentor you want ?". We had no clear preferences, but answered: "Anybody, except Mentor A" (he was local, he was not from the field we have targeted, etc). But later, by chance, we have met Mentor A, and he was sooooo valuable, he helped us to avoid some serious mistakes :)
That does not mean that you should not have any criteria. You should. Try to make the filter of 3-5 MAIN criteria. Try to search examples of his/her consultantions. Check the LinkedIn or just "google".
With "sustainable" mentorships - you are not going to marry mentor. Some mentors just help you solve one little problem and you move forward. But if you want something more sustainable then: a) never lie b) allways give feedback how you have used the advise and what was the outcome (I allways love that feeling when somebody remembers you after 2 or 3 years and attributes the impact to your mentoring session c) do the homework if you agree on something. Mentor will never be doing things instead of you. And even if paid he will be frustrated if you will be comming to the same question/situation day after day without any progress from your side.
Good luck !
I've been pretty lucky to have great mentors on my start up journey.
The best way to find great mentors is to make a short list of your ideal mentors and contact them. Let them know you are looking for a mentor and want to build a long term relationship. If they agree to meet you and there is chemistry, you should be well on your way to a long and sustainable relationship.
The most important thing to keep in mind is that your mentors are giving up valuable time. If you go in to your mentor sessions with a know it all attitude and don't listen to their coaching, i'm afraid you won't have a sustainable relationship. Don't forget you are there to learn from them, and in the ideal situation, they will learn from you.