No, I think that's it; it could be saying that the nieces and nephews and Neanderthals are with the cousins, lying about the family tree; or that they are "with" (contained within, a part of) the family tree about which the cousins are lying.

The original quote mentions three linguistic ambiguities, which I'm guessing are "lie about" (tell falsehoods about/are in a prone position around), "with" (as above), and "annoying" (adjective modifying unliked cousins/verb the speaker does not like doing).

Since it was spoken and not written in the story, I do also think that the lack of punctuation provides another: "Neanderthals" might not be the end of the sentence. "Our cousins lie about the family tree, with nieces and nephews, and Neanderthals we do not like annoying cousins." As in, the Neanderthals whom we don't like annoy cousins, and along with nieces and nephews, they are a part of the family tree that our cousins lie about.