It's much more common to meditate with your eyes closed, though eyes open can be a good antidote for sleepiness.
The trouble with the eyes half open idea is that it takes a bit of attention to maintain the eyelids just right, attention that would probably be better put on the meditation object.
"Whenever we feel that we are definitely right, so much so that we refuse to open up to anything or anybody else, right there we are wrong. It becomes wrong view. When suffering arises, where does it arise from? The cause is wrong view, the fruit of that being suffering. If it was right view it wouldn't cause suffering." - Ajahn Chah
"Remember you dont meditate to get anything, but to get rid of things. We do it, not with desire, but with letting go. If you want anything, you wont find it." - Ajahn Chah