Hi Peter, well, given that every second person in Dolwyddelan is named Williams it's not likely there's a connection, but I'll be glad to try

In the photo is Moses Williams, born 1863 in Dolwyddelan, married to Catherine Jane Jones in 1887 at Llanrwst (dist.), and died in Patagonia in 1918.
Son of Robert Williams, born 1831/32 in Llanrwst (dist.), married to Judith abt 1860, died 1905 in Llanrwst (dist.).
I do not have proof positive of Robert's parentage, but I do have an original marriage certificate of Moses and Catherine Jane. Robert Williams signed as a witness (the "original" is not the actual, signed, certificate but a clerk's copy) and the name on the signature line is actually Robert Robert Williams. So I believe his father must have been Robert also, but have no details. There was a Robert Williams born 1796 in Dolwyddelan according to the IGI, son of William Griffith and Ann.
That's about all I've got for you. Oh, and the Moses Williams family lived at Ty Capel.
Cheers,
China