There have been studies done with healthy infants whose ankles were tied to a mobile with a ribbon (the earliest of these is cited below; it has been oft-repeated with additional variables). The reward of the mobile moving produced more leg-kicking. This confirms not only that infants watch mobiles, but also that they attempt to interact with mobiles and that they remember simple cause and effect. Infants tested 1-3 days after their first training with a mobile remembered the mobile and immediately interacted with it. A 3-minute reminder session would alleviate forgetting after a 4-week interval of not being with the mobile. Researchers also found that if you introduce a new element to a 5-element mobile or moved the mobile to a different environment (new background, different smells or lighting), memory is reduced.
So beyond providing stimulation, mobiles appear to aid in the development of memory and the understanding of cause and effect!
Rovee-Collier CK, Gekoski MJ. The economics
of infancy: a review of conjugate
reinforcement. In: Reese HW, Lipsitt LP,
eds. Advances in Child Development
and Behavior, Vol 13. New York, NY:
Academic Press; 1979:195–255.