Why do you think that your Jane
must be named after a relative?
I think you might be trying too hard to make this all fit.
The names Jane and Genevieve have different origins. Genevieve is French, Jane is English. Yes, families can and do have nicknames seemingly unrelated to their given names and can use non standard short forms. There could be any number of reasons why Genevieve was known as “Jane” ... maybe she felt that the name Jane made her ‘fit in’ better, maybe she simply didn’t like the name Genevieve?
Depending on how often you have found her referred to as “Jane” this may also have been a one off error or misinterpretation written on a document. I’m sure we’ve all seen names botched on ‘official’ documents.
“Jennie” sounds like a logical pet name for Genevieve I would say.

Sweeping statement ... I don’t think Jennie would be a common pet form of Jane.

Jeanne is a French form of Jane, perhaps this could sound similar to Jennie depending how you say it?

You might find a loose connection that way? There are no rules though.

The names Jane and Patrick (and Ryan) were so common unfortunately without further information, looking at names alone I don’t know how you can find out which branch of Ryans they belong to.