There are some kinds of half-shut eyes, here are some of them:
1) While a person is very tired and he is about to sleep, or alternatively before death or if having a structure of such eyes. It can be called simply "half-shut eyes" or "half closed eyes"
I saw his half-shut eyes. (reference)
N.B. Sometimes it may be called: piggy-eyed or "pink eye".
2) While suspecting in something or having a concern or a thought about something: half-shut eyelids or "half closed eyes".
"A woman at the information desk with half-shut eyelids managed to
tell me that the KLM flight was delayed with no ETA." (reference)
The officer looked at him narrowly through half-closed eyes... (reference)
3) While a person want to improve his eyesight (by changing the shape of our eye and letting in a limited amount of light that is more easily focused) : squint or less common blink (see definition No.1)
Why can people see more clearly when they squint their eyes? (reference)
4) While conveying a glance: "bedroom eyes"
Marilyn Monroe's signature look was her bedroom eyes. (reference)
That half-closed 'bedroom eyes' look actually makes women think you
are SHIFTY, not sexy. (reference)
5) Pathological disability to close the eyes completely (for example due to a malfunction in a facial nerve): Lagophthalmos (or Nocturnal lagophthalmos if it is at night)
He has Lagophthalmos. He can't close his eyes completely.
So in the end of the day, the choice depends on context. But in your case I would consider the option No.2 here.