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Lisburn records may be online but I do not think they go back to the early 1700's which is when your William was born.
However, there may be probates, taxes and odd census. I enclosed a link to records in Northern Ireland, plus a couple links. Was he a Presbytarian or the Quaker Hancocks?
http://www.proni.gov.uk/guide_to_church_records.pdf
http://www.discovernorthernireland.com/Public-Record-Office-of-Northern-Ireland-PRONI-Belfast-P3031
https://books.google.com/books?id=a2TlAgAAQBAJ&pg=PA182&lpg=PA182&dq=lisburn+ireland+Proni&source=bl&ots=zHZg7SjQ1p&sig=hH28-j1yRrAYxLbnmabmrQV5YLA&hl=en&sa=X&ved=0CCoQ6AEwAzgKahUKEwjh6q-4mqnIAhVCbj4KHZe-Bak#v=onepage&q=lisburn%20ireland%20Proni&f=false
http://gene.pelan.org/surname-pelan/hancock
http://boards.ancestry.com/surnames.hancock/2651/mb.ashx