As noted in Ch. 21 - "Sir John wanted the whole family to walk to the Park directly and look at his guests. Benevolent, philanthropic man! It was painful to him even to keep a third cousin to himself."

If there was any actual kinship, I doubt it was even as much as a true "third cousin." I agree however that it would be just as likely and as easy for Lucy to drop the right names at the right points of conversation to make an entirely imaginary blood connection seem like folk you ought to have been in correspondence with since childhood. If Lucy so much as said she went to assemblies with someone who had become sister-in-law to an actual third cousin or so, Mrs. Jennings might well have conclusion-jumped from those names to a cousinship. She was remarkably quick at finding things out, after all.