This is a family with a series of interesting names, too many to cover in a single post.
It should be stated, first of all, that the father's name was George Washington Branson. George was born in 1832, a hundred years after the original George's birth. So there was clearly a history of applying larger awareness to the child's name in this family.
In 1851, George and family headed west from Illinois to Oregon. There he married Mary Eliza Wood. They homesteaded in Yamhill County. But don't think that just because they were living on the edge of civilization as they knew it, they were less aware of the world around them. If anything, their isolation may have heightened the cultural ties they were determined to make for their children.
Henry Clay Branson, b. 1859 was the couple's first child.
There were at least 2 children named after Henry Clay and born in Oregon: Henry Clay Branson and Henry Clay Huston, 1857-1901. Both were named after this man, Henry Clay.
Henry Clay (1777-1852) was a congressman from Kentucky, the Secretary of State from 1825-1829, and a renowned orator. He was the author of the Missouri Compromise and a founder of the Whig party. In 1844, he ran for president against James Polk. Polk's campaign slogan "54.40 or Fight" proposed fighting Britain for control of the Oregon Territory to the 54th Parallel, above Vancouver Island. The stance appealed to the voters in the East, and Polk won the election in part based on this issue.
But the people in Oregon, who would have been conscripted to fight for this slogan, may have felt differently. I can't find any children named after Polk in the Oregon Territory. Clay proposed seeking a compromise that would avoid possible war with Britain. This was the compromise that President Polk reached in 1846, giving the area between the 49th and 54th parallels to Britain.
In 1857, when Henry Clay Huston was born, Oregon held a constitutional convention. And Henry Clay Branson was born almost exactly a month after Oregon became a state. These families must have been tremendously relieved that a war had been avoided. These Henry Clays, born five or more years after the death of the original, were tributes to the peaceful Oregon that he had proposed.