Besides some of the songs mentioned already, the only "novelty" songs that pop into my head are from the 1960s, Here are some examples:

The Name Game written by Shirley Ellis & Lincoln Chase, and released in late 1964.

http://www.youtube.com/watch?v=5MJLi5_dyn0
The Name Game-Shirley Ellis; YouTube clip

Ellis also had a hit song in the mid 1960s "The Clapping Game, That song was based on two children's handclap rhymes {"3 6 9 the goose drank wine" and "my mother told me if I was goody"}. But, for whatever reason, I don't remember this song at all. However, I definitely remember Shirley Ellis' other big hit-"The Nitty Gritty" {1964}, though that's not a novelty song.

**
I also remember:

Long Tall Texan
(Henry Strzelecki)

Recorded by The Kingsmen - 1963, and others since

..."Well I'm a long tall Texan
I ride a big white horse
(He rides from Texas on a big white horse)

Well people look at me and say
Hurrah hurrah is that your horse?
(He rides from Texas on a big white horse)
Yeah"...
http://lyricsplayground.com/alpha/songs/l/longtalltexan.shtml

and Monster Mash {1962) and Alley Oop {1960)


**

Again, sorry. I know that these songs aren't from the 1950s and therefore don't fit into the specific topic of this thread. But a lot of folks like me who were kids in the 1950s liked singing them. So that should count for something.

:o)