There's no law that says you can't actually earn money under 16 and the problem stems from 'do gooders' a hundred years ago who were not entirely happy with small children sweeping chimneys and working down coal mines.
In todays labour market employers are going want you to have skills or a strong back & no brain. One of my first jobs at 16 was stacking 15kg milk crates (12hr shift, 30minutes for lunch + two 15min teabreaks) because it was cheaper to employ me than pay somebody who was qualified to run the robotic pallet stacker.
That saturday I ached. Father dragged me to the pub with him and for the first and only time I might add he bought me a pint and after telling the landlord to "sod-off he's earned it" spoke to me quietly and said "this is my life, don't let it be yours as well".
Over the years there have been many instances of people your age displaying some entrepreneurial spirit, in the mid eighties several rose to fame as writers of computer games and more recent examples can be found dotted across the news archives if you look for them. Plus there's always the slim chance somebody may want you for acting, modeling or voiceover work, but you won't know till step up and say I wanna do this!
The Internet is also worth a closer look, thanks to free packages like Wordpress and free web hosting, you don't need money or programming skills to create a simple website or run a forum like this one and cover it with adverts. You won't get rich overnight but it's a small start.