I therefore took note when I came across what seems to be a reference to one of those books Lewis gave away, which I thought I'd share here for those interested in such things.
The book in question is by Charles Williams, the first of his theological books (the first to be published, anyway): HE CAME DOWN FROM HEAVEN (1938). Alice Mary Hadfield, in her biography of Williams, notes that
On the flyleaf of C. S. Lewis's copy of He Came Down
From Heaven, in Charles's hand, is written 'At Shirreffs,
2.10, 4th July 1938'. He must have been spending a lunch
-hour with Lewis at his favourite restaurant-bar,* Shirreffs,
at the bottom of Ludgate Hill, under the railway bridge
across the road from the King Lud pub, and have given
him a copy of this new book to read in the train home.
At the end of his copy Lewis wrote 'July 26 1938',
probably the date when he finished his reading and
making notes. Sadly, Shirreffs has gone, and the site
no longer holds a restaurant
[Hadfield p. 164-165]
Hadfield's source for this is identified in a note on p. 245: 'By courtesy of Mr. George Sayer.'
If follows, then, that Sayer must have Lewis's copy of this book, and it seems likely that he was given it by Lewis himself.
As it happens, we have Sayer's own account of that event, in his biography of Lewis (JACK: C. S. LEWIS AND HIS TIMES), in which Sayer describes making as his initial pick George MacDonald's UNSPOKEN SERMONS, which Lewis was apparently too attached to to be able to let go, hence Sayer "hastily withdrew my choice and asked to be allowed to have something else" (JACK p. 249). That 'something else' now looks likely to have been the Charles Williams book -- an odd choice, I shd have thought, for Sayer. As Chuck Berry says, it just goes to show you just can't tell.
P.S.: By the way, Lewis himself left an account of this lunch-meeting in his Preface to ESSAYS PRESENTED TO CHARLES WILLIAMS (p. viii):
[During the period 1936-1939] There were many meetings
both in my rooms at Magdalen and in Williams's tiny office
at Amen House. Neither Mr. Dyson nor my brother, Major
W. H. Lewis, will forget a certain immortal lunch at Shirreff's
in 1938 (he gave me a copy of He Came Down From Heaven
and we ate kidneys 'enclosed', like the wicked man, 'in their
own fat') nor the almost Platonic discussion which followed
for about two hours in St. Paul's churchyard.