Having just come through this I thought I should share a quick recommendation for any birthing partners. My hospital let my partner stay while I was being induced, then Again for the first night after I'd given birth. However he didn't get anywhere to sleep..
If we ever did it again (which we aren't!!) we would take a blanket and a pillow for him and maybe even a camping mat. You could leave it in the car but it would have made a massive difference for him!
It's easy to forget the dad's in the process of everything but my husband was awake for 50 hours except for napping in a chair.. I was too but I survived on hormones mainly!!