A little more input regarding the name 'Arthur'.
I have noted that a child named Arthur was baptised to Roger Willis at Paul in 1713 so we know the name existed prior to the marriage of William and Margaret.
I also noticed a child Arthur baptised to 'Arthur' and Blanch Willis at Paul in 1744 making just two examples of the name in the Willis family prior to 1754. (Bearing in mind that the earliest baptisms in the OPC database for Paul are currently from 1694.)
That Roger apparently named a son Arthur points to a reasonably valid reason for William and Margaret to have used the name. Perhaps Arthur died young or tragically for example. But then we have the problem of Arthur son of Arthur and Blanch.
And just what is that problem? Well, that problem is that there is no record of a marriage between an Arthur Willis and anyone named Blanch!!!
BUT!! - PETER Willis and Blanch Keigwin, both of Paul, were married at Madron on July 13th 1734!
Zib mentioned a naming pattern in her post and I think it worth noting that now. The normal pattern would be for the first son to be named either for either the father or grandfather of the child and or the first daughter to be named for the mother or paternal grandmother. Probably more common that the first son and daughter were named after the paternal grandparents but that was not always the case.
The children of William and Margaret Willis in order were Arthur, Jane, William, John, Margaret, Thomasine, Nicholas, Mary and Elizabeth.
Arthur - unkown for the time being.
Jane - I would expect her to be named for her paternal grandmother (see Margaret below)
William - this one is obvious and he is named for his own father.
John - with two sons named before him he could have been named either for his maternal grandfather (see Margaret) or possibly for a brother of either of his parents. More likely for the grandfather.
Margaret Vingoe was baptised at Sennen 6th January 1732/3 daughter of John Vingoe and Margaret Reed so daughter Margaret is obviously named for her mother or, perhaps, her maternal grandmother BUT she is the SECOND daughter. And that then suggests the first daughter was most likely named for the paternal grandmother. And that being the case would point to the parents of William Willis being Roger Willis and Jane Crankan!
Thomasine - Roger Willis gave this name to two daughters so perhaps that is significant!!!
Nicholas - a name used by Peter Willis
Mary - being a common name and her so far down the list of children makes it difficult to speculate.
Elizabeth - same comments as for Mary.
Worth noting that Margaret Vingoe had only one sibling - her brother William baptised in 1730.
The 1744 baptism of Arthur is interesting although really only for the fact that his father was actually Peter Willis. (It was far from uncommon for the Vicar to fill out the register incorrectly by inserting the name of the father the same as that of a son. That is to say if the son, as in this case, was named Arthur then the vicar would write in 'Arthur' as the name of the father when it should have been something else. This also happened in the case of female children with the name of a daughter and the name of the mother being identically entered even though the mother's name might have been different. And there were also instances of the child's name being entered incorrectly as that of the father or mother)
But since checking further I find there was definitely at least one earlier Arthur Willis who died at Paul in 1705 and he could account for the name being used by Roger and Peter.
One thing appears certain and that is that Arthur was not the name of William Willis's father so the question is just why was that name used for his first child. It also appears certain that William did not name a son for his own father.
William son of Roger and William son of Peter were baptised just 1 year apart and we already know the age of William as recorded at his burial must be incorrect. But I am still unable to determine exactly who was the father of William Willis.
More work to be done!
CT