I personally am persuaded that John Walton's view is a rather good explanation. Essentially, the story of Eve being made from one of Adam's sides (the Hebrew word is "side," not "rib," used elsewhere in the Pentateuch to refer to one side of a building) is a vision given to Adam by God to demonstrate Eve's ontological equality with Adam. The words used here for "deep sleep" are used elsewhere in the OT to denote a dream state where God gives a prophetic vision. Showing Eve as Adam's "better half" demonstrates Eve's status as fully human, and that through marriage, the halves are rejoined to create one whole ("Therefore shall a man leave his father and his mother and cling to his wife, and the two shall become one flesh.").
This doesn't mean single people are somehow less human, but it does tell us that man and woman were made for each other. This also tells us that Eve was not necessarily specially created; she could have been born from one of the many homo sapiens that were then emerging, and chosen by God (as was Adam) as a representative. But we must also remember that more than anything, the story of Adam and Eve is archetypal, telling us things about not only them but about all of us as well.
If you'd like to read more about this particular perspective, I recommend John Walton's book, The Lost World of Adam and Eve. It offers some much-needed insights into the text of Genesis 2-4 and is all around helpful in trying to figure out what to do with the historical Adam and Eve.