For Johnny, at the beginning of the novel, the rising sun symbolizes wealth, power, and prosperity.  Johnny has a cup that has the Lyte crest on it, and he believes that he can use the cup to prove his genetic linkage to the Lyte family.  So to an early Johnny Tremain, the cup and rising sun are symbolic of his desires to rise up and be a powerful, rich, and influential member of society.  

As Johnny matures and works for the Lorne family, he begins to take on opinions of the Whigs in Boston.  Once it is revealed that the Lyte family is a Tory family, the rising sun and family crest changes meaning for Johnny.  It now represents British imperialism and oppression.  

Somewhat ironically though, the rising eye and its spreading rays symbolizing a rising of power and a spread of that power and knowledge still applies to Johnny even as a Whig.  It just represents the spread of a different political agenda.  Instead of British crown and king, it is now colonial freedom and independence.