I don't have an issue with children paying rent to their parents, but you do have to be flexible enough to work around circumstances.
For example, when I hit sixteen, I started working in a tea room, earning the princely sum of £19 a week. Gosh. Gotta love that under 18s don't qualify for minimum wage, right? My mum didn't charge me rent, but she did stop my pocket money (which had been £2.50 a week as money was always tight) and used the saving to increase what she gave to my little sister. (Don't know whether it's related or not, but my sister has always had a reputation for money burning holes in her pockets. She was slow to get a job and didn't chip in until way later than me. Even now my dad subs her, and she's living with my mum and working 'part-time'. But that's complicated.)
When I left school and got a full-time job, I started paying £33 a week, to cover a basic 'rent' and the petrol costs of picking me up from work while I learned to drive. (There was a bus there but not a convenient one back.) I payed for my own lessons and tests, although I was given a car when I passed. I had absolutely no problem with paying rent, or the fact that my mum practically dragged me to the bank to open an ISA (though I did quibble how picky she got about the calculation of my 'owings', she was fastidious to the point of ridiculousness). When I moved in with my dad, I did the same, until my health deteriorated and I was signed off work. I still chipped in by buying household supplies when I could afford it, but never returned to FT work, which I felt "guilty" for until I moved out. After all, he was still supporting my sister, and recieving no extras for me suddenly living with him and using food/electricity/water.
I take pride in the fact that everything I own I worked for (or, sadly, now recieve benefits for, sigh), something I don't think I could have learned if I'd never had to worry about other people's claim on my hard earned money.
As a comparison, my cousin was tragically orphaned aged fifteen, and spent the next few years with my grandmother, before moving to nearer ourselves to go to college. He was supported by the rent from the house he interited which had been rented out, but failed to economise, or fully realise that my grandmother was bearing most of the brunt of the mortgage on it. Since finishing college he has lived with my Dad, and doesn't contribute a penny. Well, okay, every now and then he is generous and helps by buying breakfast cereal, fizzy drinks and ginsters pasties. My dad doesn't want to bring up the subject of money after all he's been through, but there's no denying that he's going to get into one hell of a mess when he goes to Uni and is on his tod. It's just a shame that where his mother lived (and my nan lives) is out in the middle of nowhere, and there was no opportunity for him to get a taste of the real world before his bereavement.
I don't think that every child 'needs' to pay rent, but it is a very good way of gettin a taste of reality before moving out when it all happens at once. I certainly will charge my DS and his soon-to-be-sibling rent when they reach a point where I can take a tithe without leaving them short, though PT work will be different, as they will have to be in education til 18. If they earn an awful lot, they will have to contribute, but if they only work saturdays at minimum wage, then it will replace any pocket money/outting subs. I certainly don't think it will do them any good to effectively learn that they don't have to pay their way, especially since my health means I will never work FT again, so I'm hardly an example of grindstoniness.