What were Hispanic people called in the US in the 1800s?

There was originally no unified term for such people. In Texas and California the Spanish-Mexican people were called "Mexicans". The census never recognized Mexicans separately until the 1930 census, in which they were "Mexicans". The term "hispanic" is an invention of liberal elitist academics during World War I. For example, to quote from the newly minted "Hispanic American Historical Review" founded in 1918:

The term "Hispanic America" is coming more and more into regular use. It has been adopted in a number of the universities of the United States in place of "Latin America". However, the latter term still is used almost generally in government circles in this country because it was early adopted as the official title. Hence, there is a Division of Latin American Affairs in the Department of State and a Latin American Division in the Department of Commerce. It is to be hoped that these two departments will see fit one day to make the term "Hispanic America" official.

-- editors of the Hispanic American Historical Review (1922)

Chicano was originally a gang term used by Puerto Rican and Cuban immigrants in Chicago and New York. When I was a kid in the 1970s, "Chicano" implied a Caribbean native, like a Puerto Rican, Dominican or Cuban.

The hispanic concept does not predate the 1980s. It was constructed in the 70s by bureacrats and activist groups and made official in 1980.

Before then, people of Spanish speaking origin were various different nationality groups that were unrelated. They were classified by their nationality and their race. Most were considered white.

There was no such umbrella term in common use in the 1800s that corresponds directly to our "Hispanic" or "Latino" category. I think T.E.D.’s answer is correct in that people with Mexican origins were called Mexicans. But for the most part, there was no distinct “Hispanic” identity, and so there was no need for an umbrella term. Mexican-Americans in the Southwest thought of themselves as Mexican-Americans, Cuban-Americans in Florida and Louisiana thought of themselves as Cuban-American, and Puerto Ricans as Puerto Ricans. This jives with jfrankcarr’s comment: People would use “Cubans” or “Mexicans” based on whichever ethnic group they were most familiar with, or based on their assumption about the background of the person in question. So if you lived in an area where most people who have Spanish names were of Mexican descent, then residents may have assumed that a Cuban strolling through town was a “Mexican.”

So when did Americans begin to think that it made sense to lump Mexican-Americans, Cuban-Americans, and Puerto Ricans together into one ethnic group? As Cristina Mora argues in a recent book and journal article, the emergence of the Hispanic identity is a recent historical process spanning the 1970s and 1980s, which was due to 1) increased immigration to the US by non-Mexican Spanish-speaking nationalities, 2) the government trying to make society more “legible” by classifying several communities together, 3) activists in the various communities realizing that they would be more powerful as “Hispanics” than as many different groups, and 4) the development of national media markets, so that companies realized it was more profitable to market to “Hispanics” than to many different groups.

Mora writes:

As late as 1969, the U.S. Census Bureau classified Mexican Americans, Puerto Ricans, and Cuban Americans, the nation’s three largest Latin American groups at the time, as white, effectively lumping their information in with data on Anglo Americans. A third-generation Mexican American, for example, would be classified in the same category as a person of Irish descent. By the mid-1980s, however, the Bureau had instituted a new Hispanic category that sorted persons of Latin American descent into their own panethnic classification. (ASR)

And it wasn’t just that the Census was behind the times. People of Latin American decent similarly did not embrace a panethnic identity:

Indeed, differences among the various Latin American ethnicities were so vast that attempts to unite them in the early 1970s often failed. For example, a prominent Spanish-speaking political unity conference held in 1971 disintegrated as Puerto Ricans accused Mexicans of trying to impose their agenda on them, and both groups questioned whether they had anything in common with Cubans. During the same period, former census officials recall conducting focus groups on the East Coast and in the Southwest and finding that Mexicans, Puerto Ricans, and Cubans did not identify panethnically: “People didn’t even know what Hispanic meant!” recalled one official. (ASR)

But from Mora's book, today: “Hispanics in the United States represent different national origins, skin colors, socioeconomic classes, regions, generational statuses, and even languages.”

What happened in the interim? From the book:

A profound shift occurred in American history between 1960 and 1990. During this period, federal agencies developed a separate Hispanic category that effectively lumped together all Latin American communities. At the same time, large Mexican American activist organizations began courting Puerto Rican and Cuban American constituencies in an effort to develop the nation’s first panethnic, Hispanic political advocacy groups. Additionally, media executives began connecting Spanish-language television stations across the country to one another, forging a national “Hispanic” network that reached Mexican America, Puerto Rican, and Cuban American audiences alike.”

So the organizations that were key to creating a “Hispanic” identity were the Cabinet Committee on Opportunities for Spanish Speaking People, the Census Bureau, the National Council of La Raza (a social movement), and Univision Communications Corporation (a media organization). These four organizations often coordinated their efforts to create the panethnic Hispanic identity, even exchanging staff to some extent.

To get social scientific for a second, ethnic groups are not pre-existing categories, but are always constructed through political, social, and cultural processes. Put another way, neither Mexican-Americans nor Anglo-Americans in the 19th century saw a need to talk about “Hispanics” as a single class. Cuban-Americans, after all, were likely to respond differently to the Mexican-American war than would Mexicans and Mexican-Americans. Because large scale Puerto Rican immigration and Cuban immigration to the US are primarily 20th century phenomena, there was no need for a national discourse around "Hispanic" issues and so that category did not emerge as a useful or meaningful term in national discourse. (As jfrankcarr notes, Catholicism was a hot issue in the 19th century, so "papist" was a socially meaningful term, but this covered many non-Hispanic ethnic groups.) Similar stories can be told about the creation of a pan-Asian racial group – and the pan-European "White" racial group as well.

As an aside, RI Swamp Yankee’s comment is probably correct, although the use of “Spanish” as a category may be regionally specific. Mora notes that "Hispanic" had some rival terms--Raza, Spanish Surname, Spanish, Latino, and Spanish-Speaking--"but none was as popular as Hispanic was between 1960 and 1990.” New Englanders may have gravitated toward “Spanish” as a general term because New England's large Portuguese and Brazilian communities make the Spanish/Portuguese more salient there.