Benjamin Franklin used the pen name Silence Dogood, posing as a middle aged widow. Was 'Silence' a common name at the time? If not, why did he use such an unusual name?

I can't speak to the commonality of the name "Silence," but I do know a fair amount of about early American newspaper culture.

It was extremely common to write letters to newspaper editors under pseudonym, and those pseudonyms were often tropes (allegory, metaphor, etc.). The Federalist papers, for example, were published under "Publius," a Roman first name with ties to the concept "the people," (for more on classical allusions in pseudonyms during the early republic, see Eran Shalev, "Ancient Masks, American Fathers: Classical Pseudonyms During the American Revolution and Early Republic," Journal of the Early Republic, 23 (2003): 151-172.

Personification was common in pseudonyms as well. In a case I studied for my dissertation, which dealt with the presidential pardoning power, authors wrote under names such as "Mercy," "Hope," "Justice," and "Justicia," in addition to names like "M." and "AB." The allusions were not always very clever, as you can see. In Dogood #4, for example, Franklin writes about two porters named Riches and Poverty who stand at the gates of what is later revealed to be Harvard.

As for Silence Dogood, scholars have argued that the name is a direct reference to contemporary events. George Horner, for example, argues that the name could only be understood in light of the various controversies playing out in Bostonian culture in the 1720s. He writes:

The pseudonym, "Silence Dogood," is ironic allusion to contemporary events. The possible derivation of Dogood from Cotton Mathers' Essays to Do Good, a book mentioned in [Franklin's] Autobiography, has been noted. But the phrase do good had common use in Boston. As early as 1710, Benjamin Wadsworth published An Essay to Do Good by a disswaisive from Tavern-haunting and Ezcessive Drinking. More to the point is Cotton Mather's March spoken to, published in 1721, which bore the sub-title, A Brief Essay to do Good unto the Widow; Dispensing those Lessons of Piety which are the Portion Assigned for the Widow in the House and Word of God. Franklin apparently used a stock phrase to indicate that laymen as well as ministers had something to say on right action. Indeed, one tenant of the Couranteers' creed was that the ministers of Boston, far from "doing good" by their exhortation and denunciation, did harm. One of the Courtaneers complained that the ministers "who are generally of a Contrary Party to the Bulk of the People with respect to the public Affairs here in this Town," incited the people to wrong conduct by meddling in political matters and by exhortations to "peacable lives," and against mutinous action. Even Samuel Sewall held this opinion at one time. The contrast between the mildness of the Widow Dogood's correction and the vehemence of the ministerial attack must have had ironic meaning to the contemporary Bostonian, and the name "Silence" for one who set out to preach against the "vices of the town" enforced it." (George F. Horner, Studies in Philology, 37 (1940): 501-523, at 509-510.)

This seems like a relatively reasonable interpretation to me.