In further searching the NY Newspaper Death Notices, I came across one from 1866 that reads:
Aug 19 John Thomas Toner and his sister Julia on 20th children of Richard of So Bklyn
Then I went to the Brooklyn Eagle itself, and searched through the paper for that date, and found this:
It reads: Toner - On the 19th of August, of cholera, James Thomas Toner, and his sister Julia, on the 20th inst. The funeral will take place from the residence of their father, Richard Toner, corner of Van Brunt and Tremont streets, South Brooklyn, this afternoon at 4 o'clock.
It appears there was a transcription error in the one I first posted, and that Julia's brother was named James, not John.
So it seems that - again, assuming that this Richard Toner family we've been tracking lately is the Richard Toner family of our Julia Toner - that the Julia in that family who was 20 years too old to be our Julia was not, actually, our Julia. It was not too uncommon to name a child after an older sibling who had died, and we can only assume that our Julia, born several years after her oldest sister died, was named after the earlier Julia. This James Thomas Toner, though, wasn't on the 1860 census. If he died in 1866, and hadn't yet been born in 1860, he had to have been 6 or younger. Julia was 9 in 1860, and so she would have died at about age 15. Imagine losing two of your children - the oldest and the youngest, it seems, a 15 year old girl and a 5 year old boy - within a day of each other.
If I could only find the Toners on the 1880 census, we could possibly confirm some of these assumptions. If the Toner family - clearly the same Toner family - were to show up on in 1880 with a daughter Julia, 8-12 years old, we'd know for sure that our Julia was named after her older sister.