Personal recommendation is always best, however, sometimes parents are not prepared to share details of "their" tutors, because they get worried their own child will loose the advantage! I tutor full time and also employ other tutors to work for me too. Approximately 90% of our work is from recommendation.
During quieter times of year, I do have a profile on the likes of Tutor Hunt, but due to my previous working history, I am very google-able. Many parents therefore first see me on Tutor Hunt and then google me to find my details elsewhere on the web, so avoiding the £20 fee....
If you go past the first page on a Google search, you are also likely to find tutor's own websites, rather than the large agency sites.
Personally, I'd avoid using older students (who may not approach it as professionally as someone doing it full-time), but use someone who holds appropriate qualifications and preferably is registered with The Tutor's Association, which independently checks qualifications, experience and client references.
Do be very clear about your expectations of the tutor, though. I always agree early on what the aims/goals of tuition are and regularly assess progress against these goals. I also provide regular (monthly) written reports on our sessions, which parents can share with teachers, if they wish, so that I can work collaboratively with the school.