Because you have no idea how much power there is in a relationship. When you've been mentoring someone for a long time, you develop a concern for that person that rivals family. In fact, Paul clearly thought of Timothy as his own son. When one keeps the relationship properly within the confines of one's own gender, a father-son relationship is as deep as one gets, and that's definitely not shallow! However, if one were to cross over to mentoring, or being mentored by, the opposite gender, then that care and concern can become dangerous.
In fact, I'd venture to say that the care and concern which is natural and acceptable in a same-gender mentoring partnership WILL become dangerous in a mixed-gendered situation. How can one person work that closely with another person without becoming attached? It would be nearly impossible.The last thing we want to have happen in a godly outreach is to damage integrity and healthy relationships (think breaking up a marriage!).
And that's why you can't mentor someone of the opposite gender.