Dear 100 Hour Board,
Was it common in Joseph Smith's time to have a son other than the oldest be named after the father? All the people I know that are named after their father have always been the oldest.
Nope. There were set naming patterns that people did back in the olden days depending on where people lived. You can read about it here or here or here. Nowadays people tend to break these naming traditions, and I'm personally okay with it.
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I was looking at Joseph Smith Jr.'s FamilyTree and he's the fourth son of Joseph Smith Sr. and Lucy Mack. Joseph Smith Jr. continues that naming pattern and names his 4th son Joseph Smith III. But then Joseph Smith III shakes it up and names his first son Joseph Arthur Smith (but no "fourth"). Joseph Arthur Smith died as an infant so he was unable to pass on the name. Joseph Smith III had multiple wives and multiple sons but didn't reuse the name. Carl gave me the insight that since Joseph Smith III knew about the Plan of Salvation, he knew that he would see Joseph Arthur again and didn't want to have two sons named Joseph in the spirit world.
Fun Fact: In the South (where I specialize in genealogy) the southerners would often name their children after people they admire. For example, I have an ancestor named George Washington -surname-. I find that to be quite happy!
My husband's name has been passed down to the second son in the family for generations (which seems to take a lot of trust that you'll have a second son). So it happens, but probably isn't common because, again, what if you never have a second son?
In order to fully research this answer I had my husband look on Family Search going back to 1635 to trace his name, and apparently his name used to be given to every single son in the family as their first name, and then they had different middle names. I don't know if that was a common practice in Germany at the time, but at least it's what his family did. Over time this morphed into just naming the first son with the traditional family name, and then it looks like a few generations actually chose to not use the family name at all. Eventually someone brought it back, but gave it to their second son, and since then it's mostly been a second son name. It hasn't been passed down incredibly consistently, though, and keeps skipping generations; nobody in his father's generation got the name, so my husband's dad chose to bring it back when he had his second son (my husband).