I suspect the names may have been surnames (last names) of other relatives, pressed into service as first or middle names. Nothing wrong with that. Again, the idea is to both honor the descendent and to try to imprint on the character of the child. "Prescott" is not a normal Christian saint name, but it was the last name of a Bush family patriarch. It's the middle name of George P Bush.
SD
No, in the case of these country whites, the names were actually confected out of whole cloth, chosen for the way they sounded or for similar reasons, created as "one-off" namings for an individual child. One can speculate as to why they did this, but it is not uncommon in some parts of the rural South to encounter this sort of naming "convention" (or unconvention, if you will). It's seen in Texas; I don't know about other states, but suspect it's a Scots-Irish thing and would not be surprised to hear it's found in states that originally supplied Scots-Irish settlers to Texas: Kentucky, Arkansas, Tennessee, Missouri, northern Louisiana, the North Carolina piedmont and mountain counties, western Virginia.
Using last names of favored ancestors or friends of the family is more a feature of the "deferential society" of the Atlantic slope, especially among people of English backgrounds. Yankee brahmins like the Bushes would certainly be among that number.