RORY: This is one ugly looking baby. Whose baby is this?
LORELAI: That’s your second cousin’s Stan’s. Poor kid.
RORY: Ugh, he got Stan’s everything.
If Stan is Rory’s second cousin, then surely he is the son of Lorelai’s cousin? But where did Lorelai get a cousin from? We’re only told of one aunt, Hopie, who is Emily’s sister and lives in Paris (no husband or children were mentioned). It is implied that Richard is an only child, despite the proliferation of Gilmore aunts and uncles. And could Lorelai’s cousin already be old enough to be a grandfather?
It’s possible that Lorelai (or the writer?) means that Stan is Rory’s cousin twice removed, meaning that he is Richard’s cousin – if so, he’s presumably a lot younger than Richard since he’s just welcomed a new baby. Many people get second cousins and cousins twice removed mixed up.
The baby doesn’t actually seem particularly ugly, to me it just looks like a normal baby.