Nicholson was a free agent after leaving Newcastle during the summer.
"He's a young lad who we like the look of," Grayson told BBC Radio Lancashire.
"He trained with us a little bit last year and he's been with us all through pre-season. One or two going out has released some funds for us to sign Alex on a full-time basis.
"Hopefully he'll develop into a good footballer and be part of the Preston squad for a long time to come."
Preston face Swindon Town on Saturday.