As a little twist to the movie idea, if there is a drive-in movie in your area, these are always a lot of fun, and much more private (since you're still in your car--or my personal favorite, if you own a truck and put an air mattress in the back and bring lots of blankets!), and is not "stuck at home." And then if you want to keep with the '50s vibe, go out for malts/milkshakes at a late-night diner, and since it's late it wouldn't be so crowded. Also, you can take them to-go, eat them together at a park, or whatever.

Long, long, long drives, are just always a lot of fun, if you're into that kind of thing. Especially that part where you get lost, or you almost drive off a mountain or something.

Most bowling places are ghost towns in the middle of the day, unless there's a tournament going on. So maybe call and check ahead. I think a lot of different places are kind of like this--really really crowded during peak hours, but completely empty during off hours. Ice skating/roller skating rinks also tends to be that way...

Hiking. Can't get much cheaper or more solitary than that.

Maybe I will come back if I think of other things...