I know he had a son also called William Jackson
As it's such a common name you are going to need more specific info.
Have you got William Jnr's birth cert. ?
This will give mother's maiden name, you can then look for the marriage which will give William Snr's father's name.
I'm guessing he was born in Cumberland as he doesn't seemed to have moved very far.
There are 2 or 3 that might be him all born around the area he lived.
Although he died in Whitehaven he was buried in Moresby so there may be others of his family there also.