Updated Christmas 2015, original post from 2009 and updated in 2013
Shops in London are certainly open on Christmas Eve (for example the Apple Store (5pm) and Hamleys (6pm) on Regent Street, and John Lewis (5.30pm) on Oxford Street. Larger supermarkets will also be open too and they usually have a pretty good selection of toys, DVDs, stationery, and all those little gift cards (eg iTunes, Debenham's etc).
But if you're not able to get out to shops and want to send something electronically (or print to hand to someone if you have a printer) here are some ideas.
If you've got time to get some flour, salt and paint you can make salt dough decorations and if your shop also has eggs, sugar and flavourings you can make edible decorations or cakes etc.
Gingerbread men (and women) - if you don't have a cookie cutter you can make a template from paper and cut your own. Another recipe for gingerbread biscuits.
From Londonist - How to make your own tube roundel Christmas tree decoration (salt dough and paint)
Amazon giftcards / gift vouchers
There are options for sending something by email or printing. Don't forget that if you'd normally give someone a £10 book you might need to give them a £15 voucher so they don't have to pay extra for postage and packaging (hat tip @philbradley).
Cinema tickets, music tickets etc - note that this works best if you're attending with the person as you may need to show the ticket office the card you use to book. If you want to sort things out so that this step isn't needed it's probably best to ring the booking office, though admittedly they might not be open on Christmas Eve.
iTunes vouchers for music (iTunes store) or apps (app store) - I am not sure if you need to have iTunes on your computer though or if you can do it via other shops, eg Amazon. I believe that if you and your recipient both have iPhones (which are Apple products) you can even send them apps and voucher codes by phone, but I've never tried it.
People might enjoy being a member of an event or cinema venue (eg Southbank, BFI). Or gyms if people are that way inclined. Or perhaps classes for things like cookery or photography.
Lend with Care (from CARE)
http://www.lendwithcare.org/gifts/ - charitable microcredit sort of thing, see also Kiva
Toilet Twinning (donate to projects to build loos to improve sanitary conditions)
Printable Christmas crafts including 3D Christmas snowman and tree
General Christmas printables
Christmas baubles (£2.95 to download, but there's enough info to make your own)
Christmas gift tags