I can't remember how long ahead you have to book, but going to see the Ceremony of the Keys (basically ritual locking up for the night) at the Tower of London is free - you just have to apply.
*See if you can get tickets to the houses of parliament?
*I know she may have been to lots of them, but I would still recommend a couple of hours (at least) wondering round the major museums and galleries, finding the obscure nooks and crannies - there's loads of cool stuff that is easily overlooked. (e.g. British Museum, National (Portrait) Gallery, V&A, Science Museum, Natural History Museum.)
Other things I wish I'd done (or done more of) while I was still a student in London:
* Walk the river from the Thames Barrier to Hammersmith (and indeed do the Thames Barrier visitor centre)
* Cheap tickets at the Royal Opera House
* And the National Theatre
* William Morris House.
* Other more or less obscure museums: The Estorick Collection; Wallace Collection; Courtauld Institute; Museum of Domestic Design (at Middlesex University); St Bride's Institute; Type Museum; Museum of Brands; Design Museum; Museum of London; Coram's Fields/Foundlings Museum; V&A Museum of Childhood; Sir John Soane's; National Army Museum; RAF Museum; Imperial War Museum; London Transport Museum; Jewish Museum; Temple of Mithras; National Maritime Museum; Wellcome Collection...
*Go watch the racing at Herne Hill Velodrome (free?)/Olympic Velodrome (not free?)
*Parks and Viewpoints - Greenwich, Hampstead Heath, Parliament Hill, etc.
*Pub Theatre/Comedy - check Time Out listings?
*Free concerts at the various conservatoires & music schools (and also at the music venues, such as the South Bank, Barbican, ROH &c.)
"Queue for a Prom at the Albert Hall - take a picnic.
*Go see BBC radio comedy recordings (usually free?)
*Universities & Museums often have interesting lectures/events that are cheap or free?
(Edited to add a couple more suggestions)
