Stay over
Definition 1 of 2
Pro Tip 1/3Social Gesture
Implies you're close enough to the host to stay at their place. 
I felt so at home when they said I could stay over.
Pro Tip 2/3Safety First
Used to avoid traveling late at night or in bad weather. 
It's snowing heavily; it's safer to stay over tonight.
Pro Tip 3/3Unplanned Event
Can suggest an unexpected change of plans. 
The meeting ran late, so I had to stay over.