I think I got really lucky. I was able to reserve a specific site, but my brother-in-law was not. We were both refreshing (F5) our computers as 7am approached, but for some reason I was able to reserve a site in June and he didn't. So I guess that's one trick; have others in your group trying too.
I practiced the day before: I checked out the reservation website and picked some random day and site that was available just so I can see how the site works. When the morning came, I was in and out of there super quick. Unlike Ticketmaster that holds your tickets for a couple of minutes while you decide, Recreation.gov doesn't seem to do that until you're a couple of layers into your transaction. That means someone else may be looking at the same site at the same time. Ya gotta be quick.