I saw it as David identifying with T.E. Lawrence who, in the film at least, was shown as being conflicted in his identity - and perhaps his loyalty - between the British and Arabian cultures. I imagine David felt betwixt and between, being both humanlike but not actually human. I briefly considered the masochist subtext to Lawrence's character, but...no.
Edit - Found this:
David has been watching Lawrence of Arabia while the crew of
Prometheus was in stasis for two years, why that movie?
Lindelof: Ridley and I started talking about Lawrence of Arabia, for
some reason, very early on in our process. I'm a huge David Lean fan —
we were talking about The Bridge on the River Kwai and then Peter
O'Toole etc. etc. we just started saying oh what if David was just
obsessed with Lawrence of Arabia? Why would he be obsessed with
Lawrence of Arabia, and i think the short answer was: Lawrence is a
stranger in a strange land. A white man who is entirely different,
ultimately becomes the most pivotal figure in that movie, independent
of his differences. That felt slightly analogical to what we wanted to
do with David.
Source (io9.com) - thanks to Thaddeus for posting the link in this thread