What was Big Willy Shakes going for?
In order to understand the real meaning behind the phrase "neither here nor there," we've first got to go back to the very beginning of the scene. Desdemona is stuck fast to this weird, mourning mood. She begins to sing the song about the willow, which is bad news, as willows are symbolic of disappointed love. (Remember in Hamlet, Shakespeare killed Ophelia off by having her "fall" out of a willow tree.)
So this song is essentially about a woman who makes excuses for her awful lover because she loves him so much. The woman in the song doesn't blame him at all, but when she calls him a cheating jerk, he declares that the more women he gets with, the more likely she is to seek out other men.
Desdemona and Emilia then talk about whether women are ever as awful to their men as men are to their women. Emilia is certain this is the case, especially when it comes to cheating. Desdemona asks whether Emilia would ever cheat on Iago, and Emilia tells her that plenty of women cheat, and you could justify cheating in lots of different ways.
So when Emilia brushes off Desdemona's question so easily, she's really trying to get Desdemona out of this sad mood she's in. But she's also trying to get her to see the light and realize what's really going on. Emilia doesn't know the whole story, but she does know that Desdemona is more trusting and honest than Othello thinks she is. And that worries her a lot. A whole lot.
Shakespeare came up with the friction between these two women's worldviews in this scene, but he wasn't the first say this phrase. It was already popular in Shakespeare's day. In 1583, it was recorded with a note "as they say" meaning it was already something people went around saying. Looks like someone beat him to the punch on this one.
The Bard tested the waters with the phrase in The Merry Wives of Windsor. But word really got around when he wrote it in this play. It might be because this is such a famous scene.