Everyone's eyes don't do this when they're flat out with a cold?
You know how (I'm assuming) you have difficulty focusing (vision) when you're really sick? So it becomes difficult to read, pay attention to movies, drive, etc?
That's because your eyes are controlled by muscles (direction and irises) and when you're super sick, you body don't send energy to your muscles (including your eyes). It darn near shuts off your somatic motor system (the part of your brain and nerves that control your conscious muscles) so it can shunt any and all availble energy to your immune system. ((It also shuts off your digestive system... those two systems, digestion and movement use the most energy. It's not physically possible for your immune system to run at high speed AND motor core and digestion to also be functioning. Actually, half of "feeling sick" is your immune system.))
"Bleary eyed" is even a symptom of BEING very sick. Your irises don't contract and dilate as they should, and both eyes don't track sharply and together. It's ONE of the reasons doctors have you follow their finger with your eyes when you're sick... it's part of the diagnostic process to test how lethargic you really ARE.
This only happens when you're really, REALLY sick. When you're on the verge (or have passed) into TRUE lethargy (barely able to move / rag doll limp).
To know... medically speaking, when your kids get 'rag doll limp' (lift and arm and it just flops back down... that's the time to head to the ped office for same day appt, or ER if they don't have an appt. It may or may NOT be accompanied by a fever, but is a sign of massive infection.
We all say we're "feeling lethargic" when we slow down and the 10 feet to the bathroom is just soooooooo far (and so is the 6 inches to the remote!), but when kids (or adults) actually go limp like a noodle and are UNABLE to sit, walk, move about... it's ER time.
You're probably not there, and may well not get there, but it's a good thing to know as a parent that it bumps and illness from "nasty cold" to "see doctor immediately". Like "104 and out the door". Rag doll limp or true lethargic don't rhyme with anything cool, but it means the same thing as 104 and out the door. Doctor time!!!
But the eye thing happens a LOT with illnesses that don't get to doctor time. It just means really, really sick. If the rest of their body follow suit... then it's time to carry them to the car and head to the ER.