Flanagin rhymes with:
melugin, beghin, burgin, coggin, duggin, fagin, feigin, goggin, googin, hangin', holguin, jungin, kagin, koryagin, kosygin, mangin, melugin, mungin, muskegon, nagin, olguin, riggin, rogin, scoggin, scogin, scroggin, singin', wergin, wiggin, zalygin
Flanagin sounds like:
fillingame, fillingham, fillingim, flamson, flanagan, flanigan, flannagan, flannigan, flenniken, flinchum