If you wish to see the context for this post, please look
here. In short, of the three existing Fanch Mysteries - those of Fanny's birth, of Charles's death, and of Daniel's existence - this relates to the first.
Last night, while searching the family papers for something else entirely, I came across a handwritten family tree. The papers include several such, made by different hands at different times, often for different branches, and I've not been through all of them yet in detail. Anyway, on this one I saw the following record of Charles William Butler's marriage:

"Frances Niel ch. of a Frances Giberne b. c. 1740 + a Mr Niel." These are rich (if fragmentary) pickings. Until now, all I knew of Fanny's mother was that she was probably called Frances, going by Fanny's own baptismal record, which records her parents as "Clotworthy and Frances Oneile."

The name "Giberne" unlocks many doors, for it was the maiden name of Charles's own mother, Anne. And, among Anne's numerous siblings, there was indeed a Frances, born in May 1741 - which counts as circa 1740 in my book.
Wait, you may say, surely there are other possibilities - and other Frances Gibernes? Conceivably - but Giberne was an unsual name. They were a Huguenot family, and though they were notably prolific they had only been in the country since the 1680s. More importantly, for Fanny to have been taken in by the Butlers, it makes sense for there to be a close family connection. Being the child of Anne's younger sister (they were three years apart) is very plausible; indeed, it hits the Goldilocks spot. Any more distant and the obligation to look after Fanny might have seemed too weak; any closer and the laws of consanguinity would have prevented Charles and Fanny's later marriage. (As it was, they were first cousins.)
We have a few facts about Frances Giberne. She was married on 6 August 1771 to John Dyer in St George's Hanover Square, with Weeden I officiating. (This was the same church were Fanny would be baptised eight years later.) John Dyer was a native of Wedmore in Somerset, and at some point after 1773 (when their son George was born in London) they moved back to his ancestral turf, which is where both Frances and John later died and were buried.
There's not much else to go on, but living in Somerset does put Frances Dyer within the orbit of Clotworthy O'Neil, then a merchant banker in Bristol. What seems mostly like is that Frances and Clotworthy had an affair, a pregnancy ensued, and Frances was taken to London for the lying in and baptism.
What happened after that? It seems that Frances continued living with John, and Clotworthy with his own wife Mary (nee Arundell). Who knew what, and when, and how did they all feel about it? This of course I cannot tell. It would help if I could find out who brought Fanny up, and where - but there is currently no trace of her between her baptism in 1779 and her appearance in a letter of September 1798, by which time she appears to be quite at home - a fixture, indeed - at the Butler breakfast table in Cheyne Walk.
There are a few possibilities:
a) She went to Weeden and Anne as soon as she was born and spent her entire childhood with them. This is possible, but it's odd there should be no earlier mention of her in that case (not that documents from the period are very plentiful). And, if Charles grew up with her as a kind of younger sister throughout both their childhoods, is it likely that the later romantic attachment between them would have taken place? (Both Mansfield Park and Wuthering Heights assure me in their different ways that it is possible, however...)
b) She stayed with John and Frances Dyer, with John accepting her as his own, until either Frances's death (in 1786) or John's (in April 1798), and at that point moved to her Butler relatives in London. This is certainly plausible, and the fact that John died just a few months before the 1798 letter mentioned above is striking. That was also the year when George Lemon Dyer, Frances and John's son (and Fanny's older half brother, on this theory) got married and perhaps left home. (Side-note: George would include "Weeden" among the names he gave to one of his own sons, which suggests that there were no hard feelings towards the Butlers.)
c) She was taken in by Clotworthy and his wife Mary, and raised as a younger half-sister to their daughter Phyllis, at any rate until Clotworthy's death (date unknown) or Mary's (1793). She certainly seems to have been on very good terms with Phyllis (Mrs Yeo) throughout their lives. And, of course, she used the surname O'Neill.
Other solutions are possible, and of course the truth may involve some combination of the above - Fanny's early life may well have been rather peripatetic.
This discovery also narrows slightly the possible interpretations of the letter quoted at the end of
this entry, in which Fanny is discussing her (probable) son Daniel: “There is a letter somewhere, written to my Uncle, which is of consequence, because there in he acknowledges him." There is no one on the Giberne side who would be a plausible candidate for the Uncle referred to here. So, it seems likely that Fanny is referring to Clotworthy's relatives - possibly back in Ireland (for Clotworthy seems have been a second-generation immigrant). Finding that person, however, will probably be like finding an O'Neill in a haystack.
There are still many questions, in short - but at least I've inched forward a little, which I had almost despaired of.