Hi Paul,
I'm afraid I have more resources for Anglesey than for Caernarvonshire (and Lazarus Williams is a much easier name to research than William Jones!). What I have been able to find is William Jones and Catherine still living in Bangor in 1911 (William 31 b. Bangor, Catherine 27 b. Llandegfan).
Based on this information, and William's mother being Jane, there is a family in Bangor in 1881, 1891 and 1901 which may be William's, although I have no proof. Do you have a copy of William and Catherine's marriage certificate? It might give some supporting evidence.
1881 - RG11/5576/91/8 - at 5 Glanadda Terrace
William Jones, 34, slate dresser, b. Bangor
Jane, 26, b. Conway
Thomas John, 6, b. Bangor
Mary, 4, b. Bangor
William, 1, b. Bangor
1891 - RG12/4667/81/42 - at 15 Glanadda Terrace
Jane Jones, 36, widow, charwoman, b. Conway
Thomas J. , 16, slate worker, b. Bangor
Mary, 14, b. Bangor
William, 11 (transcribed 17), b. Bangor
Richard, 8, b. Bangor
Solomon, 8, b. Bangor
1901 - RG13/5279/127/23 - at 244 Carnarvon Road
Jane Jones, 46, widow, b. Conway
William, 21, general labourer, b. Bangor
Richard, 18, carpenter house, b. Bangor
Solomon, 13(?), general labourer, b. Bangor
Regards