According to the Vascologist Joseba Lakarra, Basque gorri 'red' derives from a lexeme *gorr- 'raw meat'. However, the isolationist paradigm of academic Vascology precludes further investigation of cognates in other language families.
Considering the voiced initial stop in Basque was originally voiceless and also the trill rhotic isn't the product of gemination, the most likely candidates would be Altaic *kʰjú:ru ‘red, reddish; brown, dark’ (EDAL 1090) and IE *kreuH₂- ‘blood, gore’ (Greek kréas 'meat', Lithuanian kraûjas 'blood'), thus pointing to a lexeme *kʰruH-1 ~ *kʰuHr- which would date back to the Upper Palaeolithic and underwent a semantic shift from 'raw meat; blood' to 'red' in Altaic and Basque.
One interesting derivative of gorri is gorringo2, which refers either to a kind of mushroom (Amanita caesaria) or the egg's yolk.
1 IE *e is the ablaut vowel and doesn't play any role in the actual etymology.
2 With the variants gorrinko in the Easternmost dialects (Roncalese, Zuberoan) and korrinko in the extinct Alavese.