The point of surnames is that they don't change--that they leave families easily identifiable to the state and others. So when we discount "Ellis Island names" and cases where people changed their names on purpose to obscure their heritage, I don't think there's any particularly reason to expect European Americans to have a significant number of wholly . There were lots of people who purposefully changed their names after Ellis Island, however, especially Jews. My name, for example, is very WASP-y looking, but it was a conscious attempt by my grandfather to hide that we were Jews, though he did this in Germany in the 1920's before emigrating to the U.S. after the rise of the Nazis. This was changing of last names was a big deal into the 1950's. Witness the dueling editorials (both written by Jews) in the Atlantic in 1948: "I Changed My Name" and "I've Kept My Name". These were mostly not new names (except for a few, like Mirrenoff to Mirren, which I think you'd consider "Ellis Island misspelling") because the point of the practice was to assimilate better, not stand out.

I have a longer post on how last names "work" in general (it's my second most upvoted post): As wives traditionally take their husbands' surnames, does that mean there are fewer surnames than in the past?. The point is for them not to change.

So, while there might be occasional changes, we shouldn't expect very many changes. I am not positive of colonial history, but my strong impression is that all or nearly all British and Irish immigrants had last names, as did Huguenots, Dutch immigrants, Germans, and other early immigrants to the United States. Similarly, by the time there was mass Scandinavian and Ashkenazi Jewish immigration, those groups almost universally had adopted last names, as far as I am aware. Family names not common in the United Kingdom until the Norman Invasion but, by the 15th and 16th centuries, names in Great Britain were nearly universal (there's no clear cut off date). James C. Scott records the following episode:

A Welshman who appeared before an English judge in the early sixteenth century during the reign of Henry VIII, was asked his name. He replied, in the Welsh fashion, “Thomas Ap [son of] William, Ap Thomas, Ap Richard, Ap Hoel, Ap Evan Vaughan.” He was reprimanded by the judge to “leave the old manner . . . whereupon he after called himself Moston, according to the name of his principal house, and left that name to his posteritie.”

Swedish names, on the other hand, were crystalized fairly late--men had patronymic last names for centuries, but like the Welsh names above, they did changed every generation. Gustav Andersson's son would be Thor Gustavson because, well, he's Gustav's son. This patronymic last name wasn't formally abandoned (that is, last names didn't crystalized) in Sweden until 1901, but I believe in America many of these patronymic names became crystallized as family names fairly early in their assimilation experience (or at least, that's my impression based on Willa Cather novels). The Swedes, Danes, and Norwegians at least were not coining completely new names. So even people without hereditary family names in their home countries didn't necessarily adopt "new" family names in America.

The one case of immigrants adopting a whole new I'm aware of is an "Ellis Island" type name change, but that of adopting a name of their place of origin somehow. I only have a fictional example in mind, but I've heard of other cases where it's more than a misspelling: the Godfather, Part II, the Vito Corleone was originally "Vito Andolini", but registered his name as "Corleone" (after his hometown, Corleone) at Ellis Island. It's a new name, and not simply a misspelling. But I'm guessing that's not what you had in mind.

Where we'd expect new names is in people who didn't already have family names, or at least whose family names were ignored by the state. Africans brought to the New World in chains were eventually given new last names, but mostly modeled off existing last names--Washington and Jefferson, for example, are two American last names that are disproportionately black. Some popular new last names were coined ("Freeman", for example), and while I'm sure more unique rare ones exist, all the most common last names African Americans are also common among European Americans except for Muhammad. In the 1960's and 1970's, many Black Muslims, especially from the Nation of Islam and the Five Percent Nation, changed their last names ("slave names") and adopted new ones. Elijah Robert Poole became Elijah Muhammed. Cassius Clay became Mohammed Ali. Malcolm Little became Malcom X became Malik el-Shabazz. I can't think of any last names before the (re)emergence of Black Muslims that were not borrowed from preexisting European last names.

The only group that I can think of that coined "uniquely American" surnames in large numbers are the other group that didn't have hereditary family names: indigenous peoples of North America (Native Americans, Native Alaskans, Native Hawaiians). They did not come to the United States or the American colonies; both came to them. Many Native Americans, especially on the East Coast, were given "Christian names" which included a new (generally pre-existing European) last name, but by the time the United States reached the prairies, norms had changed, and many names that were originally given became hereditary family names. For example, John Fire Lame Deer was the grandson of a man simply known as Lame Deer. Many Native American, Alaskans, and I believe Hawaiian took preexisting names, but not clearly not all of them, some adopting names that had no prior existence. One of my favorites is Mankiller (Wilma Mankiller was a Cherokee chief). Not all these names were translated: Piestewa (as in Lori Piestewa, who died in combat in Iraq as Jessica Lynch was captured) is a transliterated Hopi word apparently meaning "water pooled on the desert by a hard rain". James C. Scott, based on the research of Daniel Littlefield and Lonnie Underhill, argues that earlier Native Americans might use a variety of surnames. He gives the example of "variously known as Freed Smith, Banani, Ni- zopitawigizik, and Fredrick Sagachekipoo". However, after the 1887 Dawes Act, which for the most part ended the communal ownership of Indian land, last names didn't change very much. See, the families (just as in the Welsh case above) had to register land and deal with the state, and therefore, were incentivized to adopt permanent family names to ease dealing with the state.

Family names simply are meant to not to change--that's their primary purpose, to allow people and especially states to track families (especially in terms of land ownership and taxes owed). Most current American families arrived in American with last name traditions already in place. Others, like the descendants of slaves and of indigenous people, either had last names assigned to them or chose a pre-existing surname. Some people purposefully changed their names upon immigration, others changed their last names later, either to hide their identity (like many Jews) or to emphasized it (like many Black Muslims).

I cannot speak strictly for the US, but, I can give you a general insight about the situation in the continent and if you're interest and can give you some more interesting info.

Without getting into the problem of constant misspelling of foreign names, Immigrant surnames can, of course, vary a lot depending on naming conventions. I'll give an easy example: Icelandic people use a patronym as surname, i.e. Icelandic naming convention consists of NAME + NAME OF THE FATHER WITH THE SUFFIX "-son"/"-dottir". Knowing that it's a completely foreign concept to most countries, most Icelandic descendants carry the patronym of their first ascendant in the Americas as their surname.

Another curious example is the one of the Arab immigrants in Latin America. As Spanish and Portuguese speakers normally have two surnames, first and second generation Arab immigrants normally used a very distinct rule to name their children: LOCAL NAME (so Spanish or Portuguese) + ARAB PATRONYM (middle name) + ARAB SURNAME; s. Brazilian Paulo Salim Maluf and Argentinean Carlos Saúl Menem. However, the following generations normally use their names according the Iberian rule - i.e. the middle name, initially an Arabic patronym started to be transmited as the surname of the mother of the child. As you see, the examples are endless.

As for a change in the surnames of the "dominant culture" of a country, Brazil is the only significant example that I can find. Some common Portuguese surnames reflect typical elements of Portuguese environment which are completely foreign to Brazilians, such as Oliveira (lit., Olive tree), Pinheiro (lit., Pine tree) or Parreira (lit., Grapevine). Thus, there was a trend of patriotism during the 19th century to make your surname "more Brazilian" and some families adopted surnames based on the Brazilian flora; s. Roberto Mangabeira Unger (Mangabeira is a typical Brazilian tree). Also, surnames reflecting Brazilian toponymy are not uncommon; s. Maranhão (state), Capibaribe (river), Paraíba (river/state), Pitanguy (city/river), etc.

Finally, in the French Caribbean, slaves commonly adopted the first name of their master as their surname. That's why patronymic surnames are much more common in the Antilles than in Metropolitan France.

Most Native Americans didn't historically have surnames, so these are relatively recent inventions. Among Navajo folks, Begay, meaning son, is one of the most common surnames. Yazzie means small, and Nez means tall.

Harjo is a Muscogee Creek surname, that is also found among Seminole and Shawnee, and means crazy.

The governor of the Chickasaw Nation's surname, Anoatubby, means enemy killer.

Surprised no one has mentioned this yet. When slavery was abolished, many African Americans created their own surnames in order to start anew. A lot of them were based on things like locations, the jobs they held, circumstances they and their family were in, etc.

Throughout the pacific colonies Polynesian surnames are very common as shiwn here in the Māori Battalion Roll. Genealogy, whakapapa in Māori, is incredibly important. Nearly all art, architecture, landscape, tattoos and oral history is based around, and embodies, this genealogy.
"Tamarau Waiari of Ngāti Koura recited 1,400 names before a 1890s Native Land Court hearing, in a dense interwoven genealogy including all living persons from a single ancestor about 20 generations earlier."
Examples of non-European surnames known on the international stage include Sir Jerry Mateparae, Governer General and former head of the armed services of New Zealand; Jonah Lomu, rugby player; Kiri Te Kanawa, opera singer; and Frank Bainimarama, self-appointed president of Fiji.