It fascinates me how white Americans seem to tout their Irish or Italian heritage over that of the far more prevalent German or English ancestry. Did this come from the legacy of colonialism and world wars? Or was it that the legacy of later immigration movements, and maybe also that of catholicism, meant that the Irish and Italian families were more likely to keep their traditions? Does it have to do with police brigades? What is it?
Ethnicity is largely a matter of choice (particularly among those of the Euro-American population in North America). I take this up in this article published three decades ago.
People of mixed heritage have the option of picking one ethnicity over another - and they often do. Among Catholic immigrants, there was a frequent marriage pattern of marrying one's own ethnic group in the first generation and then marrying outside one's ethnic group (but within the church) in the second. In the third generation, there was sometimes a matter of marrying outside the church - with yet another pool of ethnicity.
This process gives descendants a spectrum of ethnic options from which to choose. The choices made are diverse and personal, but there are some trends. Among these are the fact that people often set aside what they perceive to be mundane (or lower status) ethnicities, identifying with groups that afford more prestige or other favorable attributes.
One of the issues I take up in my article is how the Irish unified around their ethnicity as a matter of defense against what was often a dominant Protestant hierarchy. Because of this, being Irish was more than ethnic pride: it was essential to economic and social success. Other ethnicities did not necessarily afford people the same benefits, and so they were not as consistently expressed. This was true of those groups that were perceived as "lessers" or who were too few to band together for significant mutual benefit.
German ethnicity is important in communities that are largely German, but elsewhere, there was little advantage to expressing one's Germanness. An additional problem for German immigrants was the differences in religion: while many were Catholics, others were Protestants and still others were Jewish, splitting their ethnic group into factions that were often mutually exclusive.
Two world wars fought against Germany didn't help things when it came to status of German ethnicity: many German immigrants stopped speaking German in their households and even changed their names to remove obvious signs of German ethnicity.
In contrast to this, the Irish celebration of St. Patrick's Day has become an international event, where it is famously said that on that day, "everyone is Irish." While the Irish who arrived in North America in the nineteenth century were initially perceived as being "lessers," the millions who arrived became effective at banding together and using their ethnicity as a tool, expressed in the cohesiveness of their community. This beginning, combined with a widespread, popular celebration of all things Irish (let's not forget the breakfast cereal "Lucky Charms" and everything else like that in popular culture), has resulted in adopting ones Irish roots (not matter how small a piece of the pie) to be a positive thing.
Much of the positive and proud "white ethnic" subcultural identities in the US come originate in the post-Civil Rights era, not the eras when immigration occurred. Their makeup thus reflects what was seen as positive in the 1960s and 70s: a time when the Irish (in Ireland) were widely seen as sympathetic, while Germany had recently been vilified through two world wars. This helps explain the phenomena already described by itsallfolklore.
The development of white ethnic identity as a reaction to Civil Rights multiculturalism was a broad cultural and political phenomenon. New ideas of personal and group identity were sweeping US culture. In addition to positive ideas about local community and history, the "language and symbols of hardworking, self-reliant, and ultimately triumphant European immigrants" that became widespread in media were often contrasted to stereotypes of non-white people. The concept of 'white ethnics' thus had a political purpose: it facilitated a rhetorical approach to attacking non-whites that avoided the sort of blatant anti-multiculturalism that was becoming socially unacceptable. Today US culture still has concepts of "good" and "bad" immigrants, "model minorities," etc that have some roots in this phenomenon.
This history is explored in "Roots Too: White Ethnic Revival in Post–Civil Rights America" by Matthew Frye Jacobson.