Symbolism, Imagery, Allegory
The King's Power
The Great Seal of England is what we might call a double symbol. It's already a symbol in the real world, but it takes on an additional symbolic quality within the story of The Prince and the Pauper.
For hundreds of years, the Great Seal of England has been used to signify that the king gives his approval to whatever is written in a document—orders for an execution, for example. So it's essentially a symbol for the king's will and the king's power. Want something done? Stamp it with the Great Seal, and you can be sure it'll be done in a hurry.
The Key to the Puzzle
Considering that, it's no surprise that the royal court is pretty concerned by the fact that this seal is missing. Without it, the king can't give his approval sign off on major issues. In a way, the king is powerless without his seal.
Did you notice that the problem of the missing seal only begins after Edward's disappearance? Since the Great Seal is a symbol of the king's power, it's like saying that the strength of the king has disappeared—and it actually has disappeared, though no one knows it except for Tom and Edward.
So it's amazingly appropriate that Edward is restored to the throne by recovering the lost seal. He's asked: "Where lieth the Great Seal? Answer me this truly, and the riddle is unriddled; for only he that was Prince of Wales can so answer! On so trivial a thing hang a throne and a dynasty!"(32.31). But it's not really that trivial: by finding the seal, Edward's not only finding a seal; he's also finding the very thing that symbolizes everything the king stands for. So it makes sense that he is the only one who can find it.
But we will admit it: that's a lot of fuss over a little stamp.