It seems that, to an omnipotent and omnipresent being, the concept of rest would be academic. What would have happened if he hadn't rested?
As you imply, God did not need to rest. But He chose to, apparently as a pattern for us:
This is not the only time God has done something He didn't need to, to set an example for us. Another was when Jesus was baptised (Matthew 3:13-15).
God resting on the seventh day is not an indication that He was "tired" or somehow needed rest, but is just a way of expressing that He had completed His work of creation.
Many (probably mainly old earth creationists) believe that we are now living in that "seventh day", since the Genesis account never says the seventh day ended (as it does with the other 6 days). In this view, clearly God is not resting, in the sense of doing nothing--He is still active throughout human history, including the work of sending Christ, etc. He is simply resting from the work of creation.
I think God's resting on the seventh day was in part to give us an example to follow.
It is interesting to note that research over the years has shown that we human beings need time to rest and re-create. We are not created to go full-out, 24/7. The psychological literature often talks about taking breaks as a good way to reduce stress. Taking time to rest gives our body time to re-create itself. Even in weight training and body building the concept of rest appears: don't work the same muscle group every day (Weight Training 101).
Our minds also need time to rest. For many people, just setting aside a difficult problem is the key to finding the answer.
It's interesting to consider that by setting aside a special, Holy day for God, we also are given a chance to recreate ourselves in His image. It gives us a day to stop, and be washed by the water of the Word. It gives us a chance to see things in God's perspective, not our own limited view.
Another interesting thing is to look at the variations in the theme of 7 periods of rest.
This theme of rest in terms of groups of seven is not limited to just a Sabboth rest. It also is found in the ancient rules for planting fields.
As Israel matured, it forgot about giving the fields a rest every seventh year. Eventually, it caught up to them.