Hello and thank you for the inquiry. I cannot say that there is or is not a distant relation here. My father-in-law's grandfather, Benjamin F. Haefeli came from Basel, Switzerland, so I've been told. He was an only child. He married Mary Elizabeth Wirz of Paterson, NJ. I have NO INFORMATION TO DATE of his parents. I am also unable to find any records to date in my research here on ancestry regarding his immigration. I am wondering if perhaps he Americanized his first name when he came to American. I'm still working on this research.
My mother-in-law and father-in-law visited Switzerland and met a man who they thought might be a distant relative but that meeting did not provide any proof of such. However, they did learn that the name "Haefeli" in Switzerland is about as common there as "Smith" and "Jones" are in the USA. I have also learned that "Haefeli" ( and there are many spellings of it) is a common German name. I believe it is probably of German origin. I am going on the premise that since Basel is on the other side of the river from Germany, that at some point in time the "Haefeli's" crossed the river from Germany and settled in Switzerland. I hope that once I can really get into researching my husband's family that I can prove myself correct and find a family origin in Germany.
Doing research here on ancestry.com, I am sure you've noticed how the Haefeli name is spread across the US. There is a book called "The Swiss in the United States" by John Paul von Grueningen which I found here on Ancestry which is a very interesting reporting of how, when and why the Swiss immigrated to the United States. I found it fascinating. I hope you can locate it... you will have to look through the card catalog for Switzerland and narrow it down from there.
I wish you luck with your research ! If you find any interesting information on the "haefeli" name, etc..... I hope that you will share it with me. Apparently the name means "Little Pot" from what I've been told.