My son (almost 4 year old) just went thru an assessment by Heep Hong OT couple weeks ago, some of the things she tested was:
1) cut a small hole in the centre of paper and asked him to peek thru, notice which eye he used
2) gave him a bottle of juice and notice which hand he open the bottle
3) ask him to pour the juice into a cup and notice which hand he used
4) asked him to point at objects in the room, notice the hand used
5) give him ball to kick, notice which leg he used (let him do about 10 times)
6) give him a small lock and the key, which hand did he hold the key to unlock (do this 3 times)
7) give him a piece of hard meat (e.g. beef jerky) and notice which side of the mouth he chew the meat
Also whenever you give him things you put in the middle and which hand he used to grab the objects.
According to OT some kid will take much longer to determine which hand to use, a more conclusive observation can be drawn at the age of 4. My son used both left and right hand when he was 3 year old, now at almost age of 4 he is 80% right hand and 20% left hand.