What weakness does Godfrey display in Silas Marner?
Godfrey is a push-over, basically. He got involved with a woman for a brief instance and ended up getting her pregnant, marrying her, and hence destroying his possibilities with the person whom he is wished to marry, Nancy.
He lets his brother Dunsley toy him around and actually pays the consequences for his brother's actions just as long as Dunsley doesn't tell his Dad about his bad marriage choice.
He also lacks initiative. He would like to be in charge of his father's property and manage it, but he lacks the discipline and the courage. He also is very easily led, as he is following Dunsley's bad habits of drinking and gambling to the point of hurting the family finances.
This is basically the problem with Godfrey: Lack of gumption and tremendous co-dependence.