When hiking alone, I too enjoyed singing out loud, making up lyrics to favorite tunes. Not only did it help pass the time, I found that it kept me from hiking too fast. I also enjoyed just standing still and watching the world around me, particularly in the morning and evening. I used to carry a pretty substantial pair of binoculars (18oz or so), that provided hours of entertainment.
On a three week ski trip in Northern Minnesota many years ago, it was hard passing the time during the dark evening and early morning hours. I found that if I didn't give my mind something to "chew on," my imagination would run away from me. I used to play mind games that resulted in making lists. For example, I would create a scenario where I was on a plane that went down in a remote Mt Range and I could only have 5 items with me. You would be amazed at how many choices you can come up with and the reason why each item would earn it's place. I would play that with items I could bring vs items I might find on the plane.
With the current technology that is available, like an I-touch and some way of recharging the battery, I am sure that I would find myself having a hard time turning it off and going to sleep.