Get an EU Passport, Relatively Speaking
When I lived in France, it was tempting to apply for a passport from a European Union country to add to my American one. One grandfather was born in Italy, the other in Ireland—two of the more lenient countries by allowing second generation descendants to apply.
They are, however, just two of the the many countries offering options for “jus sanguinis” or “right of blood” citizenship through heritage, thus providing a valuable second passport.
An EU passport makes your travel through dozens of countries easier, especially if you want to reside somewhere in Europe long-term. You do not lose American citizenship but you gain the freedom to enter and stay without the restrictions or hassles of applying for and renewing a long-stay visa. You can even work, opening up options to live, do business, or retire there. And you can pass that additional citizenship to your children if you register them before they are 18 years old, thus providing them with more options to travel and work overseas.
Naturally, specific administrative regulations apply. (My hitch is that I never did find either grandpa’s birth certificate, a basic requirement.) You may have better luck, especially these days when sites such as Ancestry.com and 23andMe help the search.
Some EU countries that allow U.S. citizens to apply for passports through a parent (or grandparent in a few countries) include Italy, Ireland, Germany, Greece, Portugal, France, UK and more. (BTW non-European countries such as Israel, Philippines and others also offer potential if those apply to you.) The process and regulations differ considerably so contact the embassy of the country involved for specifics.