Hi Steve
I've now had the chance to do further investigations and I am satisfied that it is coincidence that the two Grandmothers had the same name.
First I checked Rose Annie Buesnel's birth certificate. She was born in St Clement on 17th Nov 1889 and her father was given as John Buesnel and her mother Ann Filleul Renouf. (The reference is St Clement births, 29th Nov 1888 to 3rd Nov 1894, page 73). Note that the mothers name is usually quoted by her maiden name, it doesn't confirm they were married. I then checked for John and Ann's marriage.
I'm always nervous about using the IGI although I agree that it can be a good pointer. In this case, the IGI stated they were married in St Clement but there was no mention and I eventually found it in St John on 3rd Nov 1867. (The reference is St John marriages, Sep 1842 to Nov 1870, page 171). John was a carpenter, his father was John Buesnel and was a farmer. Ann's father was Thomas Renouf and was also stated to be a farmer.
Ernest Renouf De Gruchy was born Dec 1892 and his father was Jean Olliver De Gruchy and his mother Ann Filleul Renouf. Jean and Ann were (strangely) married in St Clement on 22nd January 1880. (The reference is St Clement marriages, Dec 1873 to Dec 1882, page 14). The reason why I say strangely is because they were both living on the other side of the island. Jean's father was Olliver and Ann's father was Josue. They were both farmers.
Ernest had a brother and two sisters born from 1880 so the two Ann Filleul Renouf's were giving birth at the same time in different places to different husbands!!
Renouf and Filleul are quite common names in Jersey. In fact my next door neighbour is a Renouf. De Gruchy is a long standing name with strong connections to boat building and sailing, fishing off Newfoundland and agriculture.
On Wednesday by coincidence I met up with a gentleman called Guy Dixon who is a historian and a genealogist. His mother was a De Gruchy and he has written a book called "The De Gruchy's of Jersey" (along with Walter Le Quesne). In the book is mentioned Ernest and a complete family tree going back to at least 1732. I have found a copy of the book and photocopied 4 pages which I think you will find of interest. It is possible to go back even further but I forgot to do those pages! If you want a copy of the book it costs £13:85 and is obtainable from the Societe Jersiaise. Unfortunatly the image files are too big to attach so can you send me a personal email with your email address and I will send them to you externally.
Let me know if I can help further.
Stewart