It is fairly widely recorded that Aston's widow Ellinor (or Eleanor), by whom he had had no children, went on to marry Edward Butler, who became the second Viscount Galmoye on his grandfather's death in 1653; he seems to have been born around 1627 (making him in his mid-twenties in the early 1650s, almost four decades younger than his wife's first husband) and died in 1677. They had at least two sons, the older of whom, Piers Butler, born on 21 March 1652, would in turn become the third Viscount Galmoye and was created Earl of Newcastle by James II in 1692 (a Jacobite title that was not recognised by London). NB that he was born just eighteen months after the death of his mother's first husband. There is no record of Ellinor's death.
So what of Ellinor's parentage? I was surprised to discover that there are two different stories. The Jacobite Peerage says that she was the daughter of Sir Nicholas White of Leixlip, my 7xgreat-grandfather. But other sources, including a Peerage of Ireland from 1809, say that she was the daughter of Charles White, my 6xgreat-grandfather. (It's not clear when the ancestors started spelling our surname "Whyte" rather than "White" but it seems to have been some time in the eighteenth century.)
Each of them is slightly problematic as Ellinor's father. Sir Nicholas White was born around 1583, died in 1654, and had numerous other children. Two of his daughters are recorded as having married in 1635 and 1636. Another daughter married the Earl of Carlingford (who cooked eggs and buttered asparagus for Samuel Pepys one day in 1662 after bringing news of the birth of the future Queen Mary II); her second son, who inherited the title in 1677 on his father's death, was killed in action at the Battle of the Boyne in 1690.
It's no clear when Arthur Aston and Ellinor White would have got together, but he was only sent to Ireland in 1648, so I would bet that they met and married then or very soon after. If she was Sir Nicholas White's daughter, that means that she was marrying more than a decade after her sisters, when her father was in his late sixties; and then went on to have at least two children by another husband. Maybe she was a lot younger than the other siblings, but it looks a bit odd to me.
On first sight Charles White is much more plausible as a brother and much less as a father to Ellinor. He too was firmly on the Jacobite side in 1688-90, though possibly too old to fight actively; he was the governor of County Kildare and MP for Naas during the short-lived Jacobite regime. He first pops up in documentary terms chasing his father's legacy in 1670, after apparently several decades in Spanish and French service. He is also supposed to have been Sir Nicholas's fourth son, all the others dying without heirs. He died in 1695; his son John married in 1704 and died in 1741. So at first sight the idea that he had a daughter old enough to marry in 1650, more than half a century before her brother (or perhaps half-brother) seems a bit implausible.
But here's an odd thing. Charles White's first wife was called Eleanor. She was the daughter of Nicholas Barnewall, the first Viscount Kingsland (1592-1663). One of her sisters married the sixth Viscount Gormanstown (1608-1643). Another married the second earl of Fingal in 1636. She must therefore have been about the same age as Charles White. For them to have had a daughter, named after her, of marriageable age by 1650 is therefore tricky but not implausible. Her date of death is not known; it is recorded that Charles White's surviving children at his death in 1695 were all by his second wife, Mary Newcomen (who had three siblings who married in the 1670s and a sister baptised in 1662, so must have been much younger than Charles; see portraits here.)
Probably the first explanation is more plausible; young Eleanor White, the afterthought of Sir Nicholas White's large family, born in the late 1620s, stayed unmarried until she caught the eye of the gallant old Royalist commander in 1648; and after his gruesome death, ended up marrying a young nobleman who was about her age and who she had probably known from childhood, and lived happily ever after; it is merely a coincidence that her brother's first wife had the same name as her.
But there is another possibility hinted at by the relative paucity of detail: the young Charles White, born around 1610, and the younger Eleanor Barnewall, born about 1615, could have got together scandalously in about 1630, with baby Eleanor as the result; spurned by both sides, Charles regained respectability only after lengthy military service abroad, the death of his first wife and the glamorous demise of his son-in-law (and his daughter's subsequent aristocratic marriage). It's less plausible, admittedly, but perhaps more entertaining.
We'll never know.