How does money influence notions of gentility in Great Expectations by Charles Dickens?

This really has to do with the issues of class structure in Britain. In earlier centuries, there was presumed to be a fixed social hierarchy with the monarch at the summit, followed by the nobility, then the gentry, and finally commoners. Wealth was expected in most cases to follow this social hierarchy, with those on top being wealthier than those on bottom. Rich merchants or craftspeople could gradually become assimilated into the gentry by means of marriage. An alternative but quite rare way mode of moving up in class for the exceptionally talented was through education, followed by a well-regarded career (usually in the Church). As these modes of class mobility were gradual and relatively limited before the Industrial Revolution, families which rose in class had time to assimilate themselves to the manners and beliefs of upper-class culture.
With increased opportunities for great wealth to be obtained in the colonies and in manufacturing in the late eighteenth and nineteenth centuries came a decoupling of the relationships among birth, manners, and money. Magwitch is an example of one of the new rich; Mrs. Pocket and Miss Havisham, of an older social tradition of gentility. Pip is caught between these worlds and value systems.