I'm not a lawyer or accountant, so don't treat this as professional advice.
Do you repair many phones a week / month? If the amount you earn per month is relatively low (just a few phones per month) then it might not be worth the hassle. Though legally, HMRC do say you should register for any amount of income.
If you want to be entirely above the board, the most sensible route to take would be to register as a sole trader. This means that you'll be trading as a business that you personally operate. If you want to use a "Business name", you're free to do so, provided it isn't the name of another business. If you register as a sole trader, you need to fill in a self-assessment tax form each year and calculate how much tax you owe the government. If you earn less than £10,000/year (which I guess you do!), you don't need to pay any tax. You have a few legal requirements to keep all your receipts and an accurate record of the money you've earned.
This is way, way simpler than registered a limited company which involves a lot of paperwork and no small amount of additional overhead (usually including hiring an accountant to do that paperwork). Don't do that
Because you're legally a minor, you'll need to get your legal guardian to do any of this (you can't legally complete the sort of forms that are being discussed) as well as submit the tax return at the end of the year.
Aside from the legal registration stuff, it's normally making sure that you have a good business plan that will be profitable enough to live on, but if you're 15 I hope you're not in a position where it would be "risky" to not earn any money for a few months at a time.
Whatever you decide, good luck!