The key is not to ask how you can get them to be your mentor, but how you can get them to want you to be their mentee.
So check it out...convenient faux-wisdom...
1) Always talk in terms of their interests.
Everything you talk about should be in terms of the things they do and are passionate about. If you can get them talking about their problems and concerns, you now have grounds to approach them with meaningful conversation that, more important than solving their problem, lets you show them how insightful you are. INSIGHT is more important than SOLUTIONS. Creative thinking is highly valued in the beginning because half the time you don't have the built-up mutual respect to make you rproposed solution seem legit.
2) DO NOT BE A GERM (pronounced gurm | n: one who is likely to freak out if they meet someone they admire.)
Highly important here. Your heroes are regular people who have worked hard and reached a point of acclaim in their industry. Pay them a compliment to get it out of your system and then value their time by earnestly learning from their stories and being an engaging conversationalist. There is nothing more awkward and disrespectful than spending your entire coffee date sopping over their achievements and live-tweeting everything they say.
3) Stay on their radar
This is a fine art. Walking this line is tantamount to balancing on fishing line stretched across the Grand Canyon. The key to this is the practice and development of high social intelligence.
Social Intelligence says that, in social interactions, we can become more acutely aware of the ''vibe' or 'temperature' of the conversation, and based on this observation, can make decisions as to when to engage the other party again based on that 'temperature.' So basically, PAY ATTENTION to how they react to the conversation and use that to determine if you have an open door. It's harder than you think :).
4) Help them develop Psychological Distance
Psychological Distance is a practice that many leaders use maintain autonomy from their subordinates. It's the establishment of a clear hierarchy, clearly putting the mentor-to-be in a superiority position. If you want to reel in a mentor, help them feel some autonomy and authority. Make sure they feel that you respect their position and that they are clearly the superior and the 'guru.'
I know it sounds simple, but after a few respectful coffees full of insightful conversations that clearly carries undertones of respectful deference from you and kind authority from them, broaching the topic of you shadowing them in their job is not inappropriate. If you have skills to offer, you may be able to make their lives easier. Once again though, check the temperature of the room. It takes awhile, but if they keep having coffee, and it gets less awkward, that's a great sign.
Servant Leadership. Permission Marketing. Create a Vacuum.