In the December, 1873 issue of ST. NICHOLAS, a magazine aimed at children in the age range 5 to 18, a C. C. Haskins published an article about the horrible things that were happening to birds, and asked the readers to become Bird Defenders. The children responded, and by the following May, the letter column of the magazine was featuring notes from children who had taken the pledge to defend the birds.
So here we have letters and notes from Americans (mostly) who were born 1856-69. In some cases the magazine published their full names, and sometimes their town of residence. In a few cases, they gave full street address. While some of these children had common names, in many cases their names were unusual, or even unique, making this a possible way to identify them as individuals.
This may not be the missing information that will break down your brick wall, but it certainly adds some data you can't get anywhere else. You will know that this person was reading, and writing letters. In some cases, the letters give extensive information about their families, or their school chums.
ST NICHOLAS was published monthly between late 1873 and the mid-1930's. All the issues before 1922 are online at the Hathi Trust and can be read page by page.
Here's an example, from May of 1874 (page 437 of Volume 1, part 2):
As you can see, young Robbie Prather of Huntsville, Madison, Arkansas gives the names of his three younger brothers, and those of his teacher and many of his schoolmates.
A web search for Robert Hiram Prather will turn up other genealogical information about this person.