What symbolic power do the names Young Goodman Brown and Faith carry in puritan New England and beyond? What might the wood symbolize?
Ah, fun questions.
"Faith" is the easiest symbol. Brown's wife's name is faith, as in, religious faith. When he cries out at the climax of the story to prevent her from being tainted by the demonic rites, he is trying to save both Faith the person and his own faith (belief).
"Goodman" means just that: a good man. Brown is a good man walking on the trail through life, encountering visions and temptations and both trying to save his faith and be saved by it.
The woods are slightly more complicated. There's a long association of the trails with life's path or progress, and of the woods with the source of things dark and threatening. For example, Dante's Divine Comedy starts with Dante on the "road" or path of his life, but in a dark and threatening woods. So, there would have been that association, which had been around for hundreds of years.
Also, though, the European Christians associated the woods with the Native Americans—and the non-Christian natives with the devil. Therefore, the woods would be doubly demonic for Goodman Brown.