I've done a solo before, and am up for another one in a few months... and sometimes BFF and I go together, which is almost like a solo trip since BFF and I love to do the same things.
Some things I enjoy on a solo trip:
Riding what I want, when I want, as many times as I want. If you hit a "good" time hour or so, you can ride the Matterhorn 4-5 times using the single rider line. (Some times it's not so quick, other times, I only wait 5 minutes!)
Watch shows that normally I wouldn't slow down for if I was trying to get my rides, plus DD's rides, plus DH's rides: Great Moments with Mr. Lincoln, It's Tough to Be a Bug, Muppets 3D, whatever show is at the Golden Horseshoe, Turtle Talk with Crush.
People watching. I love coming up with silly stories about the people I see...
Eating snacks continually instead of eating meals... or, seeing how many snacks I can eat in a day... or how many new foods I can try in a day.
Riding a new ride....
Window shopping...