The Chicago franchise started itself in 1876 when they were known as the Chicago White Stockings. They remained the White Stockings until 1889 and were known as the Colts from 1890-1897. They were then known as the Orphans from 1898-1902. Of course, they have been known as the Cubs from 1903 to this current season.
The team now known as the Atlanta Braves is the only one of the eight that has been through three cities. Some may not know the franchise started in 1876, as the Boston Red Stockings until 1882. From 1883-1906, they were known as the Boston Beaneaters and from 1907-1910, the Boston Doves. After being called the Rustlers for the 1911 season, they became the Boston Braves from 1912-1935, then again from 1941-1952. They were the Boston Bees from 1936-1951. They moved to Milwaukee for the 1953 season and were the Milwaukee Braves until 1965. Since 1966, they have been known as the Atlanta Braves.
What is now the St Louis Cardinals started in 1882 as the St Louis Brown Stockings. The next year, they were known simply as the St Louis Browns. That lasted through 1898. For the 1899 season, they were known as the Perfectos. From 1900 until this current date, they have been known as the St Louis Cardinals.
The Pittsburgh franchise has only had one other nickname. From 1882-1890, they were known as the Alleghenys. Since 1891, they have been known as the Pittsburgh Pirates.
One of the historically popular teams have been the Cincinnati Reds franchise. They had the long standing history of playing the first game in the major leagues, something that lasted until recently. Sticking with the "Stockings" theme, they were known as the Cincinnati Red Stockings from 1882-1889. From 1890-1953 and 1960 to the current time, they have been known as the Cincinnati Reds. From 1954-1959, they were known as the Cincinnati Redlegs.
The NL Philadelphia team has been around since 1883. They were known as the Philadelphia Quakers from 1883-1889 and the Philadelphia Phillies since 1890.
The last team that still exists from 1883 are the franchise that is now known as the San Francisco Giants. From 1883-1884, they were known as the New York Gothams. From 1885-1957, they were known as the New York Giants. Since 1958, this team has been the San Francisco Giants.
In the original eight, the franchise now known as the Los Angeles Dodgers started in 1884. They were then known as the Brooklyn Atlantics. From 1885-1887, they were known as the Brooklyn Grays. From 1888-1890 and 1896-1898, the team was referred to as the Brooklyn Bridegrooms. From 1891-1895, the team was simply the Brooklyn Grooms. The team was called the Brooklyn Superbas from 1899-1910 and for the 1913 season. For the 1911 andd 1912 seasons, they were known as the Brooklyn Dodgers, before becoming the Brooklyn Robins from 1914-1931. They were called the Dodgers again from 1932-1957 until they moved to LA and became the Los Angeles Dodgers as they exist today.
If the National League existed with just those eight teams, as expansion did not come until 1962, there could be two divisions. I would have put Boston, Philadelphia and the two NY teams in the same division with St Louis, Pittsburgh, Cincinnati and Chicago in the other division. Of course, division play did not exist until 1969, where there was a second series of expansion with the Padres and the Expos.