Nobu can mean "trust," "prolong or stretch," or one of the most famous Japanese restaurants in the world…and Nobu, the character, fits two of these characteristics. Guess which ones. (Hint: it's not the restaurant.)
Nobu is the man who wants to be Sayuri's danna, but she doesn't love him and she thinks he looks like a "melted candle" (16.53) because of a war injury. He has absolutely no sense of humor and only three things matter to him: "Sumo, business, and war" (17.48).
This is where prolong comes in. Talking to him for five minutes feels like forever, and Sayuri and Nobu going back and forth and back and forth (and back and forth) about whether or not he will become her danna is prolonged for the majority of the book's second half. Mameha says, "You and Nobu have an en, Sayuri, and you can't escape it" (25.61).
An en is a karmic bond, and Sayuri wishes she could escape it. Sayuri is told she's made of water; Nobu is scarred by fire. The two don't mix.
But we skipped over "trust," another translation for "Nobu." Unlike some of the book's more one-dimensional characters—for example, cruel Hatsumomo—Nobu is at least nice at times. He has principles, as we see when he won't bid on Sayuri's mizuage. And he seems to value her more for conversation and friendship than as a sex object. Even the Chairman (who sees Sayuri as a sex object) says Nobu's "Friendship is a precious thing, Sayuri. […] One mustn't throw it away" (26.29).
But this brings us to the main reasons these two don't mix. Nobu has principles; Sayuri doesn't. Nobu is offended when Sayuri takes the General as her danna, but he forgives her because she doesn't have a choice. However, she seduces the Minister to drive Nobu away, and he can't forgive that.
It may sound like we're trashing Sayuri, and maybe we are a little, but it's important to remember that Sayuri isn't allowed to have principles. Principles require free will, which geisha do not have.