Comes now (or came in 2010, anyway), an only partial explanation of a phenomenon:
“Why do we move our eyes while trying to remember? The relationship between non-visual gaze patterns and memory,” Dragana Micic, Howard Ehrlichman and Rebecca Chen, Brain and Cognition, 2010 Dec;74(3):210-24. The authors, at City University of New York (CUNY), report:
“reasons for their appearance remain obscure.”
Come more recently (came in 2012, to be precise), two of the same co-authors told a less partial part of what they think about why people move their eyes when they (the people) think:
“Why Do People Move Their Eyes When They Think?” Howard Ehrlichman and Dragana Micic, Current Directions in Psychological Science, April 2012 vol. 21 no. 2 96-100.
“On average, people move their eyes about twice as often when searching through long-term memory as they do when engaged in tasks that do not require such search. This pattern occurs when people are in face-to-face situations, when they are in the dark, and when they have their eyes closed. Because these eye movements do not appear to serve visual processing, we refer to them as ‘nonvisual’ eye movements and discuss why the eyes move during thinking that does not involve vision.”