John Henry [Version (a)]:
John Henry when he was a baby
Sittin' on his pappy's knee
Took a little hammer in his little right hand
And said "Hammer be the death of me,
Lord God, hammer be the death of me!"
"John Henry" is an old folk song, Beck probably learned it off of Woody Guthrie. I say "learned" cautiously, because Beck, after the first verse, forgets the rest. He proceeds to whistle the vocal melody a bit, before whistling "You Are My Sunshine" instead.
Beck's loose cover can be found a very informal tape made of what seems to be him and some friends singing songs at a house party or something, circa 1990.