Tim Roth - The original choice for Severus Snape. Turned the role down eventually in favour of Tim Burton's Planet of the Apes.
Don't get me wrong, Alan Rickman did an outstanding job, but I often ponder how Tim would have played Snape if he ended up taking the role.
Christian Bale - One of the actors initially considered for Sirius Black.
Got with go with Gary Oldman on this one, Bale would have been 30 and way too young to play a guy who is old enough to have a teenaged kid.
Sir Peter O' Toole - Was considered for the role of Dumbledore after Sir Richard Harris' death and before Sir Michael Gambon was cast.
Naomi Watts - It is rumoured that she was considered for the role of Narcissa Malfoy, but again, that was probably just a rumour.
Hugh Grant - The original choice for Gilderoy Lockhart.
Henry Caville - Was among the actors considered for Cedric Diggory.
Sir Christopher Lee - Was considered for Rufus Scrimgeour.
Edit 1:
Rowan Atkinson - Rumours that he was initially approached to play Lord Voldemort were probably baseless. I do think he would have made a good Arthur Weasley or Xenophilius Lovegood though.
Martin Freeman - Thanks to Damini Singal for pointing this out. Martin expressed regret that he was one of the very few British actors never to have been offered a role in the Harry Potter franchise. Though, in the producers' defence, there probably wasn't a character in the franchise which would have suited him perfectly. Professor Quirrell perhaps? Not sure.
Colin Firth - Was among the actors initially considered for Professor Quirrell for the first movie and then Remus Lupin for the third, but probably no formal offer was ever made.
Tilda Swinton - Was reportedly offered the role of Sybill Trelawney before Emma Thompson was signed.
Ian McKellen - Like Peter O' Toole, was in consideration for Albus Dumbledore after Richard Harris' death. Apparently wasn't too interested as it was 'too similar to Gandalf the Grey'.