"Heavy-eyed" is a term used to describe a person who has difficulty keeping their eyes open due to exhaustion or sleepiness. There are several other words that can be used to describe this condition. One such synonym is "drowsy," which means feeling sleepy or lethargic. Another synonym is "fatigued," which suggests a feeling of physical tiredness. "Soporific" is a more unusual synonym, which refers to something that induces drowsiness or sleepiness. "Lethargic" is another possible synonym, which suggests a lack of energy or enthusiasm. No matter which synonym is used, the message is clear: the person in question is struggling to stay awake and alert.