"Born to play" probably means something different to different people, but to me it's best exemplified when you have an existing character and you hear someone's name associated with it and you just think "YES". It's not so much about an actor transforming him or herself as it is their public persona already fitting a certain role.
So, for instance, while Heath Ledger did an amazing job and arguably even best job of playing the Joker, I wouldn't say he was 'born to play it'. That'd be something more along the lines of Jack Nicholson, which had most people nodding along in agreement and going "oh, yes, yes" when first hearing of his casting.
One example I think that never actually came to pass was Gillian Andersen as Clarice Starling. Now, I don't actually think she'd necessarily do the
best job of playing the part, but it fits so well with her resume that it almost makes
too much sense. All she'd pretty much have to do is put on a southern accent and she's good to go. In fact, she apparently lobbied for the part in
Hannibal and one of the reasons she didn't get it was her role in the X-Files was too similar.
This one might well be the best example in the thread yet. Hell, it would not surprise me if the reason they even MADE a Flinstones movie in the first place was John Goodman was around to play Fred.
Another: