The most common origin that people associate with this term is this:

Samuel Mudd was a physician who practiced medicine in Charles County, Maryland during the Civil War. He was awakened by two men at 4 a.m. on the morning of April 15, 1865. Dr. Mudd did not realize that the patient with a broken leg was the well-known actor, John Wilkes Booth; had no idea Booth had suffered the injury at Ford’s Theatre while jumping onto the stage after sending a bullet crashing into the skull of Abraham Lincoln. Dr. Mudd set John Wilkes Booth’s broken leg.  Eventually, while searching for Booth, the police discovered that Booth had made his way to Dr. Mudd’s home.  Once the police arrived at the doctors home, Booth had been long gone.  But, Dr. Mudd was arrested for aiding Booth and was convicted of conspiracy in the assasination of President Lincoln, and with haboring John Wilkes Booth and his accomplice David Herold. He and was merely a kindly country doctor, unwittingly swept up in the vengeful hysteria following the war-time assassination of the president.

Andrew Johnson eventually pardoned Samuel Mudd in 1869 and he was released from prison, but the doctor’s conviction was never overturned. Richard Mudd began a campaign to restore the good name of his grandfather, but was only partially successful. He persuaded several states to pass resolutions proclaiming Dr. Mudd’s innocence, and Presidents Carter and Reagan wrote letters affirming their belief that the doctor was blameless. Both presidents, however, said they could not officially overturn the decision of the military court, leaving the controversy, and Dr. Mudd’s reputation, to the ultimate judgment of historians.

While this is the more common association with that phrase, it is not the original source. The origin actually dates back to 1823. Its earliest known recorded instance in 1823 an is in fact based an obsolete sense of the word ‘mud’ meaning ‘a stupid twaddling fellow’

Mud – a stupid twaddling fellow

To say that one’s “name is mud” is to say that “one is discredited”.  That being said, it is really easy to see why it’s origin has been more commonly associated with Dr. Samuel Mudd.