you can save through out the game on randalls diary (located in his closet to the left of his apartment), on mya's cat if youre playing as Mya, on various notepads in the hospital or on randall's diary through out the dream worlds! you also later save the game on his diary when playing as mya !
EDIT oh wait, do you mean it wont let you save from the save points? if thats the issue, id say you might have to reinstall the game ;o; it seems like a couple people have had some weird glitches when installing the game and im not sure why, but its rare and not always the same glitch (youre the first person to have a saving issue like this)