With interracial relationships being taboo until very recently (and still persisting in some areas of the U.S.) and marriages being outlawed until relatively recent years, it seems incredibly unlikely that there was ever a rash of Whites and Native Americans mixing. Yet it seems nearly every family has a great-grandmother who was full or half Indian. With those social mores against interracial relations, how did it become so popular a story to be part American Indian? I am also interested in insight as to why the ancestor is so frequently supposed to be Cherokee and/or a "princess" or some other usually-European-colonial royal title. Thanks!
I don't have an answer to the part of your question that concerns cultural appropriation of American Indian heritage by White people. However, I would point out that one of the premises of your question is fairly flawed. It was relatively common on the frontier from the seventeenth to the nineteenth century, during the time before the massive emigration of Whites across the plains, for fur traders, explorers, and other frontier types to take Indian wives. The Whites on the frontier at this time were almost universally men, and there's the old saying that "It is a truth universally acknowledged, that a single man in possession of a good fortune must be in want of a wife." Fur traders often took Indian wives, because those were the women available to them.
Here is a brief article from the University of Nebraska on the subject that includes several good sources for further reading.
I don't have all the census figures handy, but there has been a dramatic growth in the number of Americans and Canadians claiming Native ancestry since the 1970s. Note that the greatest growth has come in people who claimed to be part-Native, rather than members of a single tribe:
http://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=1063&context=pscpapers
This is self-reported ancestry - which is subjective - and part of the increase has to do with changes in the census that allow people to identify as belonging to multiple ethnic groups.
Part of this has to do with the removal of segregation laws, which made being non-white less problematic. But I think it's also a matter of attaching a certain level of cultural authenticity to being Native American - essentially a repackaging of the Noble Savage stereotype, in which indigenous people are considered proto-environmentalist, anti-colonialist, pseudo-communalist symbols. Having a long-lost Cherokee grandmother seems so much more sexy than being just another generic white dude in the suburbs.
This article on the "Wannabe tribe" expands on these dynamics: http://blog.longreads.com/2014/08/04/why-do-people-continue-to-fashion-native-identities-out-of-thin-air/
Everyone else has provided fine answers, so I'll just add to them. Certain people are keen to prove and claim Native American heritage for economic benefit. For example, I had an Uncle that tried very hard to prove Native American ancestry to obtain college grants.
Not to say that this is the only, or even a primary motivator, but it probably helps to stimulate interests and investigation.
I may also add that in America it's very common to be curious of and open about your ancestry. People with a cent of Irish blood proudly proclaim that heritage over drinks, or on St. Patrick's day, and it's not uncommon for any American to tell you 'I'm 1/8th Polish, mostly German, with a bit of Irish and maybe some Indian thrown in'. Most of that is based on family lore and consensus rather than research, but it still stands. As attitudes towards minorities have relaxed, more people feel comfortable claiming that little rumor of family heritage.
I don't think there was a single moment or place where suddenly it became cool to claim to be part Native American, it's more likely a gradual sort of thing that's been developing over the past half century or so.
While I agree that much of the claimed "Native American" ancestry among English-speaking whites is probably hokum, the "Indian Princess" trope has a long history with a reasonable explanation.
The cultural ideas of an "indian prince/king/queen/etc" seem mostly to appear when Europeans, especially the French and English, encounter some deeply unfamiliar political structures in the Native societies of North America. Most Native societies in what is now the continental united States did not have sovereign states with fixed territories as the Europeans understood the concept. Many early accounts from New England and the mid-Atlantic misunderstood the Sachems and Sagamores to be members of some sort of hereditary high-status social group, similar to European nobility, which isn't at all correct in any meaningful sense.
Throughout the 17th and 18th centuries, European-Americans consistently misunderstood the nature of Native political system, and tended to view them as more autocratic and centralized than they really were in practice. This was one of many sources of conflict with Euro-American colonists into the 19th century. Even when both sides tried to respect the treaties in good faith, there was a conceptual gap between European and Indian ideas about property and sovereignty that ultimately proved unbridgeable.