There are 2 good areas for independent shops, the Lanes as xyx said and also the North Laine. The Lanes has a lot of antique shops, jewellery, some clothes, but the North Laine might suit you better - second hand and vintage shops, comic books, art materials, Vegetarian Shoes, house decs and interiors, great cafes and cake shops.
Look out for Get Cutie and just across the road is Snoopers Paradise. here's a map of the North Laine.
Also check out a pop-up shop called 'Knowledge' on North Street, here is an example of the kind of thing they have.
The Brighton Pavilion is really worthwhile, the rooms are so extravagant, and from mid November there will be an ice rink outside - might not be there in time for half term.
even at age 16 they will enjoy a visit to Brighton Pier!
the Brighton Comedy festival is on 10-25 October, but there are always comedy nights and shows and plenty of bands and gigs and live music throughout the year.