Does Goodman Brown's cry "My faith is gone!" in "Young Goodman Brown" have a double meaning?

Yes, Faith, the name of Goodman Brown's wife, is one of our first clues that this story can be read as an allegory, a concrete and tangible representation of an abstract idea or concept. Goodman Brown plans to leave his wife, Faith, as well as his Christian faith behind for "this one [last] night," and then he plans to "cling to her skirts and follow her to Heaven." The narrator tells us that

With this excellent resolve for the future, Goodman Brown felt himself justified in making more haste on his present evil purpose.

He purposely goes into the forest on some evil purpose with the intention to be good starting tomorrow, but this is not how faith is supposed to work; one cannot simply abandon one's faith (as Brown abandons his Faith) and then pick it back up whenever one feels so inclined. If one is going to have faith in God and commit oneself to God, then this is something a person is supposed to always be engaged in, not trying to be good only when it is convenient. Thus, when Goodman Brown cries out that his faith is gone, it does seem to have a double meaning. His wife, Faith, is gone, he believes, because she has given in to the devil, but his Christian faith is gone also. It seems, perhaps, that God has relinquished Goodman Brown since Goodman Brown found it so easy to relinquish God.

When Young Goodman Brown dreams that he hears Faith's voice in the sky above him and then sees that "something fluttered down through the air," which turns out to be a pink ribbon, his Puritan belief system collapses in a desperate cry:

'My faith is gone!' cried he after one stupefied moment.  'There is no good on earth; and sin is but a name.  Come, devil; for to thee is this world given.'

After having his world view turned upside down by seeing everyone he believes is good on the way to Satan's convocation of sinners, Goodman Brown is presented with the final proof that his faith has been misplaced.  When he cries that his faith is gone, he means that not only is his wife, Faith, gone to join Satan, but also that his own faith is finally and completely gone.

When Goodman Brown begins his journey, he believes that, no matter what he does, he can rely on his wife, Faith, to lead him to heaven:

'Well, she's a blessed angel on earth; and after this one night I'll cling to her skirts and follow her to heaven.'

After his dream in the forest, however, when he hears Faith overhead apparently on her way to Satan's meeting and then sees the irrefutable proof--Faith's pink ribbon--that she is indeed on the same journey he is, this incident is the strongest proof for Goodman Brown that everyone, no matter how outwardly "faithful," has been corrupted by Satan.  

Young Goodman Brown's loss of Faith, his wife, the only person in his life for whom he has any trust, crushes the last bit of conventional religious faith in him--he loses his wife and his faith at the same time.

What does the phrase "My Faith is gone!" mean in "Young Goodman Brown" by Nathaniel Hawthorne?

This exclamation has a double meaning.  Goodman Brown's wife's name is Faith, and he loves her dearly.  She is the picture of innocence as she bids him goodbye, with her pink ribbons blowing in the breeze.  Goodman Brown holds on to this image and his love for his wife as he traverses through the woods.  By entering the forest, he chooses to leave Faith behind, both literally and symbolically.  As he tries desperately to resist temptation, he exclaims, "With Heaven above, and Faith below, I will yet stand firm against the devil!"  Then he hears many voices he recognizes from the village, among them his own sweet wife.  He then cries, "My Faith is gone!…There is no good on earth; and sin is but a name. Come, devil! for to thee is this world given."  When he realizes that even his wife is a cohort of the Devil, he gives up not only on her, but also on his religious faith as well.