http://phonetic-blog.blogspot.com/2010/02/buttered-peas.html

This extract from the above.

On Tuesday evening I took part in a folk music session in the depths of rural Walthamstow (London E17), and this was one of the tunes we played. (I was on the melodeon, and five other people were also playing melodeon; in addition we had several fiddlers, guitarists and so on.) One of the other musicians, a fiddler, commented that the name of the tune was actually a ¡°corruption¡± of a Welsh name Pwt ar y Bys, which he claimed meant ¡°fingering exercise¡±. He said the tune was originally a fingering exercise for harpists.
I had never heard this explanation of the name Buttered Peas before. The name does perhaps need some explaining, since we have no tunes called Buttered Parsnips, Buttered Cabbage, Buttered Potatoes or the like, nor for that matter Mushy Peas or Peas in White Sauce: so why Buttered Peas?
I didn¡¯t say anything beyond expressing polite interest, because I felt I had better check. As I suspected, Welsh pwt pʊt means not ¡°exercise¡± but ¡°stump, something short¡±. Ar y bys ar ə bi¢°s, ar ə bɨ¢°s does indeed mean ¡°on the finger¡± (though that¡¯s finger in the singular: the plural is bysedd ˈbəse©£).
It¡¯s certainly true that the Welsh name for the tune is Pwt ar y Bys (here), and it means literally ¡°something short on the finge.