The 1902 Waterbury City Directory shows no Handrahan's, but 3 Hanrahan's::
John Hanrahan, emp Met Life Ins, rms 51 Center. Miss Maggie Hanrahan, emp White & Wells, bds 91 Bishop. Thomas J. Hanrahan, bartender, bds, 360 Mill.
I followed Thomas J. forward thru his WWI Draft Reg and the 1920 Waterbury Census with his wife Catherine.I checked your posting history to get more info on the Gilbert family and found Catherine, born 1878 and immigrated 1896, not to be a match with the info you have.
John was harder to track down, as it seems he didn't stay in Waterbury during the census (that I could find).You might try contacting The Waterbury Library and see if they have any directories right after 1902 as they sometimes say if people died or left the city.