As long as he's actually getting adequate sleep, it should be nothing to worry about.
My daughter started doing this when she was an infant, also at about three months old. She would half-open her eyes, seeming almost awake at times (although she was not), and she would also giggle, sob, reach out into the air for things that weren't there.
What is happening is that when they enter into REM sleep the inhibitors in the brain that keep our bodies from physically acting out our dreams aren't inhibiting as well as they do in most people. This can happen in people of any age, however it is much more noticeable in infants, as their sleep cycles are only 15-20 minutes long, as opposed to 45-60 minute adult cycles. It is also more common in infants and young children, and most do outgrow it. It is not a problem at all, unless it is so active that it wakes them. Sadly, my daughter would wake every time she began to dream because her body became so active, which led to years of sleep deprivation for her and for us. However, she did outgrow it, and she is now a healthy, happy 11-year-old.
The funny thing is that one of our dogs does it, as well. She half-opens her eyes and her legs begin to scrabble on the couch...I think she is dreaming about chasing squirrels!
It sounds like your son's situation is not unusual and very mild, and as long as he is sleeping, I wouldn't worry. When that happens, he's just dreaming!