Dear college graduates, whatever it is that you decide to do during your first 90 days in your first job, do NOT ask anyone to be your mentor. In fact, do not EVER ask anyone to be your mentor. Even typing that question makes me cringe. If you only remember one thing from the many posts I write, be it that you should see things (and yourself) from the perspective of the people you interact with.
If you had been working at a company for several years with enough success that entry-level hires viewed you as a potential role model, and a just-hired twenty-two-year-old you had had no previous interaction with asked you to be his mentor, would you accept? Would you really want to give advice to an admiring youngster you can't vouch for (potentially wasting time, effort and human capital on an incompetent nonentity, a loose cannon, or who knows what else), or would you want to be more discerning regarding the people you take under your wing?
Do not even try to find mentors before you have a track record that would make your would-be mentors want you as a mentee. (This is something PepsiCo CEO Indra Nooyi also emphasized at the 92nd Y panel I attended back in September; read my blog posts about it here and here. Mentors find mentees, not the reverse.) And when you are ready for mentoring, I would advise to avoid that big word - mentor - and instead phrase questions in terms of behavior, such as: I am facing an issue and I'd love to have your input on the way I could handle it, or: did you ever face that issue when you were in my role, I can't quite figure out what to do, would you mind talking it through with me?
Not only is the term "mentor" over-used, but it also betrays the ambition of the people who want to be mentored without saying anything about the (real) benefits the people who do the mentoring gain from the interaction. Most people like to take under their wing youngsters who remind them of themselves so that they have at some level the impression of helping their younger self, but no one wants to feel used. The word "mentor" also suggests that the mentor has achieved a "perfect" state of all-knowing wisdom that the mentee strives to achieve.
This discourages a more realistic assessment of the more senior employee: most people have weaknesses, blind spots, flaws but those who are successful in their career compensate for those in a way that delivers value to their boss and their team. It takes a particularly self-aware and self-secure mentor to point out his own weak spots to a mentee so that the mentee doesn't emulate those aspects of his personality. In some cases, the mentee may even believe that the weak spots (such as, say, a tendency toward autocracy in leadership in this day and age) brought the mentor's success about when the success happened in spite of them.
In summary, "will you be my mentor?", especially asked by someone who has just started his first job, is a naive question that smacks of amateurism and a lack of basic understanding of psychology. (Also, it doesn't allow the person being asked for a graceful way to say no.) If a more senior employee has a track record in an area where you'd like to succeed, observe him and ask him questions about the specific behaviors what you've observed. He'll be glad you've taken the time to think about it first and care to ask for his perspective. Or ask for help in brainstorming an issue, but make sure you have prepared a couple of ideas first to show you value their time and can take yourself of the "basic stuff".
If you'd like him to send opportunities your way, then what you need is a sponsor, not a mentor, so you might as well get the terminology right first, but the same issue remains: why would he send opportunities your way if he doesn't know you and can't vouch for you? Show them you'll be worth their time. Then they'll put your name in for new opportunities without you even having to ask.