Hi Nancy,
Was it Brooklyn or New York City? I see 2 Timothy's that might fit that appear in the 1855 Census. I was researching the Moriarty's of Annascaul a few weeks ago, my grandmother was a Moriarty too! In one line, they had 20 children from which has sprung over 1000 third cousins. In addition to the very rich American records, we can take a look into the Annascaul Catholic Parish records which are now online at the National Library of Ireland, at http://www.nli.ie/en/parish-register.aspx but unfortunately in this case, we will not pick up Timothy's birth as the records start a fwe decades later. Griffiths Valuation of Ballynacourty, which encompasses this area, was done in 1852 though, just prior to your immigration date. Looking there I see 3 Timothy Moriarty's, in Ballinclare, Ballintermon and Brackloon. It might take a bit more scouring around to see which of these might be your Timothy.
Jean