In order to understand why 1890 was used and not 1880, it is important to note that the there were three waves of "immigration."
The first wave was up to 1840. That is, most Americans of that year were descendants of "settlers" (including slaves) over the previous 200 years, which meant that the number of immigrants in any given year was small compared to the descendants of previous arrivals. Apart from African-American slaves, these settlers were mainly of British, French or "Dutch" descent (including some west Germans.)
Beginning in 1840, in the "middle" wave, the rate of immigration rose sharply, but this was considered mainly "desirable" immigration from Germany (including eastern Germany), Ireland, and Scandinavia; that is, from northern and western Europe. This was reflected in a the growth of the American population by 46 million from 17 million in 1840 to 63 million in 1890, with about one-third of the growth represented by "first generation" Americans.
In 1890, a third or "late" wave of immigration began, with a different mix of people, mostly from southern and eastern Europe, and also from Asia. This led to an equivalent growth of 43 million, to 106 million by 1920, with the newer group of immigrants being considered "less desirable" than the older ones.
The 1924 Act had two thrusts. The first was to cap the overall rate of immigration to 2% of the U.S. population a year (actually more like 1% because the cap was 2% of the 1890 population) to prevent immigrants from being an ever-larger proportion of the population. The second impact was to revert the immigrant mix to the "desired" ethnic (not immigration) mix of 1890. As another poster pointed out, the choice of 1890 rather than 1880 allowed politicians to cater to the "middle," mid-century immigrant groups.