The "new small talk" is a term Higgins invents in act 3 to explain Eliza's bizarre manner of speaking. In the act, Higgins arranges for Eliza to visit his mother during her time for receiving company. When she comes, she impresses everyone with her beauty and poise. Higgins has told her that she can only talk about two subjects: the weather and one's health. The comedy of the scene in part comes from the hilarious difference between Eliza's upper-class accent and the lower-class slang she uses to describe her aunt's death.
There are a couple of ways to understand the "new small talk" term. While on the surface, it would appear that Higgins is just trying to cover for Eliza, there is also a sense in which he is making fun of the sensibilities of his mother and of the upper class in general. The propriety of "small talk," or conversation about things that don't matter in the blandest terms possible, is turned on its ear by Eliza's "new" small talk, which explains away her vulgar expression by turning it into the new fashion.
Of course, Eliza's story about how she suspects her mother to have been murdered over a hat is the opposite of a "safe" topic. The truth of Eliza's family's alcoholism is, in fact, horrifying, as is her matter-of-fact acceptance of it; that her description of home life is told in perfectly accented English only highlights the disconnect between the upper and lower classes. In this sense, "the new small talk" can be understood as subversive, a darkly comic way of inserting social consciousness into upper-class discourse. Higgins's mother and her friends are of course unaware of the fact that they have been made to seem like fools, something that becomes clear when Clara declares the new small talk "delightful and quite innocent."