I think creating an itinerary to follow on a solo trip is as important (if not more) as when you're going with a group. It doesn't have to be something set in stone, just a rough idea of what you want to accomplish that day. I've found that if I go solo with no plan in mind, I think more about being by myself. If I go alone with a mission, I feel like its easier to get things done. Typically, I'll go on "Photo excursions" where I make a list of places I don't have photos of and make it my mission to get some good shots of that place.
The only time I feel "lonely" on my solo trips is typically waiting in a long line. That's the time I do some text messaging, or look back through my photos.
Overall, I say treat your solo trip as a mission to get things done, whether its certain attractions, photos, or whatever. Its like running errands in a way, but the kind of errands you LOVE to do.