This is a bit of a post-in-a-hurry; but here are a few ideas;
Mary had a Baby, yes, Lord
Mary had a Baby, yes my Lord
Mary had a Baby, yes Lord
The people keep-a-coming and the train done gone.
The first phrase, “Mary had a Baby” uses just three notes; for example C D E or G A B, as does the last phrase “and the train done gone”.
Jingle Bells needs no encouragement; being the only five-note carol I have now taught it on recorders, clarinets, keyboards, chimebars, pianos……
Christmas is coming, the geese are getting fat. Please put a penny in the old man’s hat.
If you have no pennies, a ha-penny will do,
If you have no ha-pennies then God bless you.
This uses the tune of Pease Pudding Hot; notes CDEF or GABC
And of course, good old Hot Cross Buns – seasonally altered to Hot Mince Pies
Hot Mince Pies, Hot Mince Pies. One-a-penny, two-a-penny, Hot Mince Pies.
the easy tune uses just BAG or EDC. It works as a simple round if you want to make more of it.