Garfield the cat is named after a man, and through him after a president.
The cat honors his grandfather, James A. Garfield Davis. That grandfather had been named for President James A. Garfield. Mental Floss reports that the grandfather supplied more than the name. Davis described him as a large, cantankerous, curmudgeonly man with a gruff exterior and a soft heart.
The strip came after a failure. Davis's first comic, Gnorm Gnat, ran for a few years in a local Indiana paper but was not popular. He grew up on a farm with about 25 cats, which he later drew on for the new character.
The new strip did not start out as the cat's. According to Mental Floss, it began as a feature called Jon, about the cartoonist Jon Arbuckle. The cat, who scored the punchline in every strip, was the real draw. Cartoonist T.K. Ryan told Davis, "The strip must be about the cat. Go with it."
Garfield made his first appearance in June 1978. ITV, marking his 35th birthday on 19 June 2013, reported that he was first syndicated in 41 newspapers. Mental Floss puts it at more than 2,500 newspapers and journals today.
The surname itself is much older than either man. SurnameDB says Garfield is an English surname from a lost place name. It probably described a triangular piece of land left at the corner of an open field. The Old English gara, from gar, "spear," refers to that spearhead shape. The site's earliest record of the name is a christening in London in 1581.
Not every country kept it. Davis has said that in Sweden the cat is called Gustav, and that only three countries, all in the Nordics, use another name. Mental Floss names the other two as Norway and Finland.