The verb blither is a variant of blather, meaning to talk nonsensically. Aside from the a/i distinction, the words are the same except for slight connotative differences. Both usually appear in their present-participial forms, blathering and blithering.
Blather is preferred by a wide margin in all varieties of English, although blithering is more often used in the phrase blithering idiot. Both words are pejorative, but blithering is harsher (because it’s usually followed by the obviously pejorative idiot).
Examples
After this went on for a while, Smith pulled back towards the band, still having not dignified my blathering with a single word. [NPR]
[T]here are many who continued to work and develop into old age without becoming repetitive, blithering fools. [Bloomberg]
But we seem to be drowning now under the blathering classes. Gosh, some people can talk! [Guardian]
One has only to look at science and statistics rather than Mr. Duran’s emotional blithering. [letter to Porterville Recorder]