Is there any info that could explain why the Sorcerer's/Philosopher's Stone was moved from Gringotts to Hogwarts? My guess would be "Dumbledore's intuition" but I don't recall anything that would explain why the stone was moved when it was. Did Dumbledore know Voldemort was trying to get the stone? If this is the case how did he know and why did he only move it at the beginning of the first book and not some time before that?
No canon info I'm aware of, BUT:
Dumbledore suspected Quirrell was up to no good (in DH we see Snape's memories where he'd told by Dumbledore to "keep an eye on Quirrell for me"). This means he may have been aware Voldemort was on the move.
This was the year Harry would have ventured forth from Privet Drive's protection to the Wizarding World, and become widely seen. Dumbledore may have suspected that this would draw Voldemort out, and once drawn out he'd want to use the stone.
My suspicion is that Dumbledore had been planning on hiding the stone in the mirror of Erised, which is explained at the end of the movie. Dumbledore also tells Harry during the movie that the mirror will be moved to a safer location, which I take to mean safer yet for the stone - both for where it is located and how to get the stone.