I was refereeing basketball at 11 and cleaning the local church from 12. Started babysitting at 13 and then at Target at 14y9m. It made me very much more aware of how much things cost. My parents were raising four kids, with private school fees (think today's equivalent of $25K per year per child) on teacher's salaries, so there wasn't any spare money at all. In fact, I helped my folks out with money when I was in my late teens as Dad was unemployed and they didn't have $ for mortgage payments.

My kids will be starting a part time job as soon as they can, and we have set "rules" about their income, together as a family. They will be saving at least 50% of their own income, and when they want to buy a car etc (although my DS1 wants to buy a house first!!!) I will match whatever they have saved, as a one-off gift to them. So there's a huge incentive to save money and be responsible with their own finances.

I don't think 13 is too young, although it does depend on the individual child. I find children gain a lot of maturity when given responsibilities, including for their own money. I'd be a bit careful about what sort of work they did, though. Volunteering at the library, for example, wouldn't be a bad option for a 13yo. Depends on the child.

Good luck.