Sure enough, David and Eliza (Pemberton) Finkle were not too hard to find. I figured they would be in Clay, St. Clair, Michigan.
In the household of 1880, there were six children enumerated, and in 1900, there was one more son, which made for a total of seven children. These were the only seven children ever mentioned in family obituaries, so I figured I had found them all. Curiously, at the bottom of the 1884 census record, there was a seventh child literally called "(No Name)." This child was a one-year-old baby boy. All six children from the 1880 census had been accounted for on this 1884 census record. According to the St. Clair County birth registers, the seventh child was not born until 1885. So who was this nameless baby?
I decided to double check the "Michigan Deaths, 1867-1897" database at FamilySearch.org. Perhaps I could find a death record, though I had my doubts because I was positive I had already searched this database for the Finkle surname. Lo and behold, there was a death record! It was right there in plain sight. No finagling of surnames or parents' names needed to get the right search results.
That little stinker! Wesley Finkle was there all along. According to the death record, Wesley Finkle died on 20 Nov 1887 from spinal disease (poor baby!) in Marine City, St. Clair, Michigan. He was the son of David and Eliza Finkle. And he was 4 years old, making him the right age to be the one-year-old baby boy in 1884. I'm not sure if spinal disease means spinal meningitis or spina bifida. I also wonder why he didn't have a name in 1884 if he was already a year old? I could understand if he was only a few days old. Makes me think he may have been born disabled and perhaps they didn't name him right away, knowing that he was going to die. Even so, would they really wait a year to name him? Or is it more likely the census taker and the death record are off on his age by about a year? Maybe that is more reasonable.
Lesson Learned: Don't forget to search for the death and birth records of children who were born and who also died between 1880 and 1900. Thank goodness for those of us lucky to have state census records in the intervening years.