I tried to find out more about why the fox has such a patronymic and about Gediminids in Novgorod... lots of websites made by the copy/paste technique.
One trend is that the origin of this patronymic cannot be traced. Vladimir Dahl wrote in 1862 that relating sayings, such as
Lisa Patrikeevna, to the Lithuanian duke Patrikey ... would be an 'ungrounded arbitrary act'. Recently, the State Darwin Museum wrote that the reason why this patronymic had been given to the fox was 'hidden in the fairytale past'. That's about all as concerns more or less academic opinions.
The second trend is represented by a multitude of sites by/for kindergarten/preschool educators and primary school teachers that say -
Patrikeevna is associated with a Novgorod's Lithuanian governor Patrikey who lived six (as some say) or seven (as others insist) centuries ago. I guess in a couple of decades there will be many people around convinced that Patrikey gave his name to the fox. Seems to be a case of false etymology.
[ETA: I'll try to explain why I think so. Advocates of this theory provide quotations from chronicles to prove that Patrikey who governed Novgorod was cunning and resourceful and therefore gave his name to the fox. What they do prove in fact is only that Patrikey really governed Novgorod and was really cunning and resourceful.
Discussing this with Wife, I said, 'The name Patrikey must have come here and might have become associated with the fox long before the 14th century. I wonder why.' She answered, 'Maybe because Patrikey and cunning (хитрый - khitry) sound alike.' That was it.
Often, sayings are rhymed in languages, say ve
sna-kra
sna, Pa
trik
ey sam
tret
ey in Russian (only Dahl mentions this saying that means, according to him, bad fellowship, bad help), etc., so why not kh
itry (cunning) Pa
trikey (assonance/consonance). Especially so in Russian that regards vers libre (free verse) as something slightly exotic. Therefore, consonance/rhyming may be behind why Patrikey is thought to be cunning and to have been the father of the fox.]
Confusion is added by the fact that there seem to have been two Patrikeys. One is Gediminas' grandson, a son of Narimantas (Gediminas' son) baptized Gleb, therefore known as Patrikey Narimantovich or Glebovich. The other is Gediminas' great grandson, a son of Narimantas' son Aleksandr, known as Patrikey Aleksandrovich. In some cases stories I read started with Patrikey Narimantovich/Glebovich and switched to Patrikey Aleksandrovich somewhere in the middle of narration. Some sites mention them as one person (
http://fabpedigree.com/s072/f331462.htm). Some sites mention only Aleksandrovich (
http://www.roskildehistorie.dk/stamtavl ... thauen.htm). Some sites recognize the existence of both. In any case, a person named Patrikey is said to have been the governor in Novgorod in 1382-1388 and since 1397 before moving in 1408 to Moscow.
Sorry, not very conclusive information. [ETA: Though now I hope your question's been answered in a way.]