One of my favorite bits from The Simpsons involves Krusty the clown playing King Lear. Krusty feels hamstrung by the material and decides to "punch it up on the fly."
This is how I often feel when singing songs from Children's Songbook. I know at least some people feel the same way; friends of ours have changed the words to "Pioneer Children Sang As They Walked" to better-reflect the reality of the Mormon migration. They sing something like, "They washed at streams and worked and cried / Sundays they camped and some of them died." (For our part, our family shows the reality of the journey by ending the song with about a dozen more "and walked"s than the songbook calls for.)
In our family, we've developed our own words for "Christmas Bells Are Ringing" and for "Follow the Prophet." The Christmas song we've updated for use in other holidays, and the prophet song we've updated for President Monson.
We've added several versions of "Christmas Bells," but all I can remember is the Easter on, which we've used quite a bit. "Easter bells are ringing / Hear what they say to you / Easter's not about / Eggs, sheeps, and bunnies / Eggs, sheeps, and bunnies." Of course, it is to be sung in a round.
Our most-recent lyric change was the verse of "Follow the Prophet" we added. "Tommy was a prophet, tried to burn a field / But the fire went crazy and it would not yield / Later he raised pigeons, swam laps in a pool / Then stopped a friend from killing himself because that is not cool." For supporting material, see here, here, and here.
I've heard of others singing "High on a mountain top / A monkey chased a squirrel" and "Has given me an earthly home / With parents kind of weird." We also incorporate those in our family singing.