In The Lion, the Witch and the Wardrobe, what did Aslan mean when he said "All names will soon be restored to their proper owners"?
When we consider the meaning of given quotes it is important to read them in the context of the entire novel. Aslan says this important quote in Chapter Thirteen, just after Edmund has been rescued from the hands of the White Witch. The Witch's dwarf comes to Aslan's camp demanding audience on behalf of his mistress, saying rather formally:
"The Queen of Narnia and Empress of the Lone Islands desires a safe conduct to come and speak with you..."
In response to this recitation of the titles that the Queen feels she has a right to, Mr. Beaver expresses his annoyance that she calls herself "Queen of Narnia." It is then that Aslan delivers the line you have picked out. This line indicates that Narnia is not running the way that it should be and that the natural order of events has been disrupted thanks to the White Witch and her tyranny. However, it also indicates that this is fortunately going to be just a temporary state of affairs, and that Aslan will restore order soon, restoring the "name" of Queen of Narnia to its proper owners - Susan, and of course Lucy.