Fix stripped screws in a doorframe
In order of my personal preference:

Option 1: Go Go Gadget Screws
Get longer screws that reach the framing. Don't run them in so tight as to pull the jamb out of position. All-thread screws might bite into the jamb somewhat to prevent movement. 

Option 2: A Square Peg in a Round Hole
Fill the holes with wood glue and tap in wood "dowels" shaved from a scrap block. They don't need to be perfectly round but should fit snugly. Let it dry well and re-drill with a 1/8" bit.

Option 3: Bulk Up for Success
Procure screws with the same head size but a larger shank. They're somewhat specialty, so try an old-school hardware store rather than a home improvement box. 

Option 4: See You on the Flipside
Drill and countersink the hinge holes in an opposing stagger pattern. Pilot with 1/8" bit.