The "bottom button" on the blinds needs to remain "adjustable" It is the means by which you adjust the tension on the blinds and once they won't stay up, you'll have to add more tension to that button. So, gorilla gluing it in place will set you up for sure failure when that string stretches a bit.
Alternatively you can use a larger screw (not longer, larger in diameter) or if you want a more reliable repair, you can buy threaded jacknuts at any hardware store. Remove the screw, drill the appropriate size hole where the screw was stripped, install the jacknut then use a machined screw to secure the button in place.
http://www.stanleyengineeredfastenin...readed-inserts