There seem to be lots of opinions on this, but I'd like to point out that the OP didn't ask "when are you good enough to be a DM", he asked when to start it.
Good enough is determined by the instructor signing off on your cert. Some instructors are good about this (IMHO, one who ensures you learn, know the skills, have the tacit knowledge, and observes your progress and fitness to execute the responsibilities of DM). Some aren't (e.g. motivated purely by profit or by the profit potential of being able to lead "more" divers"). It is up to the potential DMC to choose an instructor/shop/outift wisely (and they don't always do this)
As for when one should start the course, here is my $0.02 (or about 0.154, OP!)
1) Be proficient with your own skills (buoyancy, all basic OW skills, your own equipment, etc)
2) Be interested in the details of diving (e.g. the answer to the questions "why" and "how" beyond (A)OW.)
3) Be interested in sharing your knowledge with others and learning from them (students, teachers, other divers, people who are just curious)
4) Have at least a few dives in different environments (e.g. fresh water, salt water, open ocean, quarries, rivers, lakes, etc)
5) Have at least a few dives in different conditions (e.g. 100' vis, 3' vis, day, night, current, no current, shore/surf, boat, sleep, little sleep, warm, cold, shallow, deep, etc)
6) Be capable of managing stress in yourself (panic, unexpected conditions, etc)
7) Be responsible.
8) Be physically fit enough to rescue someone else.
9) Have the time to committ to diving. You don't have to always be johnny on the spot at the dive shop or at the dive site, but you should undertake this with an eye toward learning, contributing, and making progress at a reasonable pace.
This certianly isn't an exhaustive list, but it gives a number of common indicators that you are ready to start. Remember that, as a DMC, you aren't just helping others, but you are learning too. You'll spend a lot more time under water and your skills will improve even more because of it. You will come to understand why things happen and how things work. You will be privy to the business of diving, the teaching of it, and maybe even perks depending on the shop. It is a worthy undertaking if you are interested in it, but it is not (IMHO) something to do just to get another card or to claim a title.
You may notice that I don't count the number of dives a diver has among the qualifications to start the program. There are minimums set by the agencies, but I find it to be an invalid criteria from a quality perspective. I'm sure there is a higher correlation between divers with more dives and their qualification to start DM (more dives = more opportunities to have experiences/experiment/get comfortable), but it isn't a sure thing by any means. Judging a diver by the number of dives they have is like judging a diver by the c-card they carry. I have dove with DMCs who's first order of business upon descending is to silt the vis to hell and beyond because they (can't? won't?) stop their descents with their BCs rather than fins. I have also dove with brand new OW students who have a mastery of bouyancy and stress response that a dive veteran would envy. "Qualified" is a subjective judgement based on demonstrable knowledge and experience, not an objective measurement based on a certification or number.
Like I said, just my $0.02. I completed the PADI DM program last Saturday and it was a great experience. It has been very rewarding and I'm very excited to work with new divers every week!