It was common for petitioners to circulate these small forms, which, signed by representative citizens from a given congressional district, could be distributed en masse in the hopes that a bill would be sponsored and introduced. I have searched available records, and cannot find specific evidence that such a bill was introduced in the House of Representatives, but it must have been, as Mercy Ann Hall's request appears later in the Congressional Record under the heading "Petition for assistance for the heirs of Captain Hall," sponsored by one George William Allen of Hall's home state of Ohio. The debate on this bill does not seem to have survived, although it appears it was referred to the Committee on Naval Affairs, and there's a small item in the Boston Evening Transcript saying that "a number of persons interested in the case of the widow of the late Captain Hall of Polaris fame appeared before the committee" and that they discussed "whether it would be better to vote her an amount of money at once or place her on the navy pension list."
A few days later, newspapers around the country reported that "The House Committee on Naval Affairs have agreed to report favorably on giving the widow of Captain Hall, of the Polaris, a pension of forty dollars monthly, and the pay due to her husband, amounting to about $2,000." Though seemingly generous --$40 would be about $750 in today's money -- this was in fact substantially less than the pension that would be given an actual US Naval Captain, which would have been more like $130 a month (more than $2,500 today). Alas, of the subsequent lives of Hall's widow and children, there seems to be very little record, and so it's difficult to say how long they lived to enjoy this pension, or whether they thrived in future years.