I think that weekly is best to start off with, if they haven't had to be used to budgeting before.
Mine started from when they were 7 yrs old, and then it moved to monthly when they went to secondary, as a sort of progression, once they'd got used to the concept of choosing to save up for things or whether to spend it as soon as you had it, and what it felt like then to not be able to have something you want, because you spent it when you got it sort of thing.
Re the amount, it is a 'how long is a piece of string?' question.
I see some huge amounts of money mentioned on some of these threads, but I think a lot of it depends on your family budget. Not only in terms of what you can afford, but in terms of what they are used to having, when they are out and about.
If you as a family are the kind of people like me who have always taken a picnic on a day out, and taken a water bottle or flask from home, and consider a coffee out somewhere to be a real treat, then your dc won't need as much as if they have grown up with people buying them comics / magazines every week...going to the cinema every week (rather than a birthday treat) ....... eating out on a regular basis (again, rather than a 4 times a year treat) ...... buying pop or coffee whenever you are out somewhere (as opposed to taking stuff with you). Their expectations will be different.
Also, how much of their time is taken up with things like sport or Scouts or drama or dance or whatever they do ? If they are looking for things to fill their time (cinema, bowling, shopping, etc) they will want more money than if they spend all day out dancing / camping / at rugby tournaments or swimming galas.