According to a Wikipedia article (which I have no reason to doubt), the Latin (and hence the English) specific name of the tree, Ginkgo, results from a combination of folk etymology and misreading.
All the OED can tell us about the etymology of ginkgo is
[Jap., f. Chinese yinhsing silver apricot.]In Chinese characters and Hanyu pinyin this Chinese etymon would be written 銀杏 yínxìng. When the tree was introduced into Japan from China the Chinese name was borrowed into Japanese with the pronunciation ぎんなん ginnan. (The Japanese pronunciation of Chinese words and “readings” of the kanji in which they are written is a topic way beyond my knowledɡe.)
But the same Chinese characters can also be read in Japanese as ginkyō, which is where the folk etymology comes in. Apparently Engelbert Kaempfer, the first Westerner to see the species in 1690, wrote down this form, with its pronunciation, in his Amoenitates Exoticae (1712). But his y was misread as a g, and the misspelling stuck.
Our pronunciation follows the spelling, so that we call this tree ˈɡɪŋkəʊ. If it had not been for Kaempfer’s bad handwriting, we’d presumably be calling it ˈɡɪŋkiəʊ.
Because of the pronunciation, people also often misspell it as gingko.
Confusingly, there is also a Japanese word ginkō, pronounced with -ŋ-. But it means ‘bank’.
The specific part of the scientific name Ginkgo biloba transparently means ‘having two lobes’, a reference to the shape of the leaves. You’d think that it would be pronounced in English as ˌbaɪˈləʊbə, since this is what we get for the Latin prefix bi- in bisexual, bifurcation, bipolar and other words. But in practice people who talk about the supposed medical benefits of Ginkgo biloba extract generally seem to say bɪˈləʊbə.