"However, one of them will say something which causes the other to laugh and then tell me it is "a private joke" but they will then carry on the joke leaving me sitting there with no idea of what they are talking about."
Well, since they used the word 'private' I'd be inclined to respond - 'Well you obviously wish to speak privately and you can't with me here, so I'll leave you to it.' And leave. Any protests I would counter with 'Your private jokes are making me feel excluded so yes, I'd rather leave.'
Don't sit there and simmer. And don't put up with it. You may be right that she's trying to reconnect with her cousin but it's still rude and if you are friends you should feel able to tell her so. If you don't feel you can say this to her, I'd give some thought to how good friends we were.
And whilst I understand some people get a bit hesitant about driving distances, it's still indicative of her attitude. It's too far for her, but not for you. She asks you to do something she wouldn't.