His eyelids were beginning to droop (=because he was sleepy).
1 [intransitive, transitive] to hang or bend down, or to make something do this: Can you water the plants? They’re starting to droop. His eyelids were beginning to droop (=because he was sleepy).2[intransitive] to become sad or weak: Our spirits drooped as we faced the long trip home.—droop noun [singular]—droopy adjective