The abstract of a fascinating 1995 review paper by Maria Casagrande and colleagues which gathered experimental data together to try and work out which of the brain’s cortical hemispheres falls asleep first.
It turns out, it’s the left.
Which hemisphere falls asleep first?
Neuropsychologia, 33(7), 815-22.
Casagrande M, Violani C, De Gennaro L, Braibanti P, Bertini M.
Behavioral tasks (reaction times to acoustic stimuli and finger tapping tasks) performed by normal subjects when sleepy or attempting to fall asleep have been used as indices of hemispheric asymmetries during the sleep onset period. Results show a stronger impairment of the left hemisphere (right hand) both in reacting to external stimuli and in sustaining endogenous motor programs. The left hemisphere seems to fall asleep earlier than the right hemisphere.
Link to abstract of scientific paper.