Marriage of Arthur Thompson and Rebecca Bell was registered in Lisburn 1847, Volume 6, page 617. Birth of Henry Arthur Thompson registered Belfast 1868 Volume 6, page 417. Birth of Fanny Thompson Belfast 1865 Volume 6, page 320. Possible death for Fanny Thompson Belfast 1865 volume 11, page 175.
The certificates will give you addresses for further research, and the marriage cert should give fathers names and the couples addresses at the time they married.