I too have relatives who said they were related to Rutherford B. Hayes, the president. My relations are Sylvester Adams Hays, born about 1827 in New Hampshire, who married a Mary E. Their son, William B. Hayes, was born about 1854 in New Hampshire. He married about 1875 to Millie J. Unknown. If I knew her name (she emigrated from Sweden), I could fill in a lot of blanks.
Anyway, Millie and William Hayes had a son, Harry C. Hayes, my ggrandfather, born in New Hampshire. Harry married Samantha Joann Wallace and had a daughter, Anna Wilhelmina Hayes, my grandmother.