I'm 25 and having my mam as a named driver still knocks £50 off my premium, even though shes never say in the driving seat of any of my cars in the 7 and a bit years I've been driving. Make sure whichever parent or grandparent or whatever is named has plenty of no claims and no recorded convictions or accidents regardless of whether there was a pay out or penalty points awarded in the last 5 years else the difference will be lower.
The black box is also a great way of bringing down the price if the kid is boring and doesn't mind being teased, bullied and left behind by mates in their cars.
The car makes a massive difference to the price. Stay away from any popular boy racer cars such as 106's, saxo's, civics, clio's, fiestas etc as they are higher risk in the age group due to the amount these cars are involved in accidents by drivers in the age group (17-24). They are higher risk. Try to buy as new a car as you can afford as better safety features help bring the price down. Look for a car less common among younger drivers such as a skoda octavia or an Audi A4 as they will generally be cheaper to insure, at least they were when I was in my first few years of driving.
My first car was a 1 litre 5 door corsa, it cost £1800 to insure, it lasted 3 months or so before I melted the clutch and shagged the gearbox, I upgraded to a 1.6 DOHC Astra with twice the BHP and got 50% of the remaining cost back as a refund. At the end of the year I swapped it for a 1.8 litre 306 XSI with even more power (150bhp if I remember right) which cost me £1057 to insure for the full year. In comparison, far less powerful 1.2 litre 106's, saxo's, corsa's etc would have cost me £1700-1800 to insure for a year.