Could Prince William run for and be elected Prime Minister and still
retain his title to the throne? If so, why?  If not, why not?

No, the House of Commons, to which the Prime Minister must be elected,
is just that, a bunch of commoners.  A titled person by inheritance
must give up his or her title to stand for election.  This has
happened, the title passing on to the next in line, but I can't
remember names.  I believe Churchill refused to accept a title to
remain a member of the Commons.  Also Life Titles, which do not pass
on to descendents, preclude membership in the Commons, which is why
Lady Thatcher, and many others receive such titles only after they
have retired from active politics in the House of Commons.  But then
they join the House of Lords, although they must not actively
participate, just as many hereditary "lords" don't bother with doing
so.
You didn't ask  - and this is only a comment -  but I believe that
anyone who is a possible heir to the throne is precluded from sitting
in the House of Lords.
The list is long.  Magna Carta and all that.

I believe Myoarin is correct, if the PM must be an MP (it wasn't
always so). For example, Tony Benn gave up his title in order to serve
as an MP in the House of Commons.

I don't think this is correct.
The short answer is that (at least in theory) the Queen can invite any
member of either the House of Lords or the House of Commons to form a
government. As Prince William is not currently a member of either
House, he would need to be elected as an MP before he could become
Prime Minister.
The Prime Minister does not have to be a member of the House of
Commons, although the last Prime Minister to sit in the House of Lords
was Lord Salisbury, who left office in 1902. The Earl of Home
disclaimed his peerage on becoming Prime Minister in 1963 and
successfully fought a by-election.
Those hereditary peers who do not sit in the House of Lords, following
the House of Lords Act are now entitled to vote and stand in
parliamentary elections.
This category would include the Prince of Wales, as well as the Dukes
of Edinburgh, Gloucester and Kent, all of whom were peers in the own
right and were entitled to sit in the House of Lords until the passage
of the House of Lords Act 1999. In the case of Prince William, he has
never held a peerage and so there would not seem to be any formal bar
to him standing for Parliament.
A full list of the offices that would disqualify someone from being
elected to the House of Commons can be found in Schedule I to the
House of Commons Disqualification Act 1975 (as subsequently amended).
This list includes various members of the judiciary, the armed forces,
the police force, certain civil servants and members of foreign
parliaments - but not members of the royal family.
Of course this would almost certainly never happen as senior members
of the royal family choose not to exercise their democratic rights due
to their close relationship to the monarch. It is understood that when
he was a student, Prince Charles had to be "advised" by the Queen that
he should not join Cambridge Universitity's branch of the Labour
Party.

On a separate point, myoarin is also wrong to suggest that life peers
cannot participate in the work of the House of Lords. In fact the
opposite is true - life peers are now normally appointed specifically
because of the contribution that they can make to the House of Lords.
Some even serve as Government ministers.
Hereditary peers on the other hand can no longer sit in the House of
Lords, with the exception of a) 92 peers who were elected to represent
hereditary peers in the House of Lords until the new structure of the
House of Lords could be agreed; b) a number of hereditary peers who
were given life peerages to allow them to continue to sit in the House
of Lords.

Bluemikewood-ga,
Your comment is very interesting and so authoritative that I accept it
without reservation.
Just one correction on your last paragraph:  I said that Life Peers
could not sit in Commons  - only in Lords.  But we agree on that.
Myoarin