I'm not a historian but I am a meteorologist who loves this sub and night be able to help.
The United States Weather Bureau was established on February 9th 1870. The first official duty if the Bureau was to take weather observations. However there were issues on which department would administer the Bureau and how weather records could be communicated. At first it was part of the Department of War, then on 1890 became a civilian based organization on the Department of Agriculture.
From my readings it seems the reason that most records start in the 1880s is because that is when the Bureau had built up enough infrastructure. Grice's research indicates that only 24 reporting stations existed in 1870 to 284 stations in 1878. Originally these stations were coastal sites on the Atlantic, Gulf of Mexico, and the Great Lakes. The stations then spread to the interior parts of the US.
I am not sure on which exact stations existed before 1880, but in 1877 the war department issued an official order on when exactly weather observations should be taken, and the first records that match today's type of weather obs practices were born.
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