One could say that the blue cross in Chesterton's story symbolizes the triumph of what the author regards as the one true faith—that is to say, Roman Catholicism—over the forces of evil.
The cross is a material object, and a very beautiful material object at that: a valuable silver cross studded with sapphires. To an international thief and serial con artist like Flambeau, this is all that matters. The spiritual significance of the cross and what it represents mean absolutely nothing to him at all.
But to Father Brown and other Roman Catholics, the blue cross is so much more than a valuable objet d'art; it carries with it deep symbolic significance. As such, its value is beyond any material worth.
In the clash between Father Brown and Flambeau over the blue cross, we see a confrontation between two radically different and incompatible worldviews. On the one hand, we have the spiritual worldview of the Catholic church. On the other, we have the purely materialistic outlook of Flambeau.
Although the blue cross is both a material object and a spiritual icon, it is the latter aspect that is the most important and the one that eventually prevails in the story. When Flambeau is finally captured by the police, we witness a triumph of the spiritual over the material, as well as a victorious conclusion to all of Father Brown's ace detective work.