It's a made-up word. It has no meaning.
From Muggles' Guide to Harry Potter/Magic/Blibbering Humdinger at Wikibooks:
No description of the Blibbering Humdinger is ever given to us. From the fact that only Luna Lovegood seems to believe in its existence, we can safely assume that it springs from the fertile imagination of Xeno Lovegood, her father.
The Blibbering Humdinger is first mentioned in Harry Potter and the Order of the Phoenix when Luna Lovegood says that people used to think that they didn't exist. Hermione replies that they were right, because they didn't exist, causing Luna to slouch off, annoyed.
In Harry Potter and the Deathly Hallows, after the Battle of Hogwarts, Luna, seeing how tired Harry is, distracts the people around herself and Harry Great Hall by saying, "Ooh, look, a Blibbering Humdinger!" This allows Harry to don his Invisibility Cloak, and collect Ron and Hermione without anyone noticing.
Although it has no meaning, it might have been a play on words intended to convey the same kind of meaning as gibbering or blithering, which it sounds like a combination of. From both gibber and blither, respectively:
: to speak rapidly, inarticulately, and often foolishly
: to talk foolishly or nonsensically : BLATHER
But that's purely speculation on my part.
Hermione's comment is mainly meant as a snide comment, referring back to her statement that no such creature exists. (But the syntax makes it sound as if blibbering is something that a humdinger does.)