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When Neo sees the Oracle, she tells him that he's not the One. She also told Morpheus he would find "The One" and told Trinity she would fall in love with The One.

Later Morpheus tells Neo that The Oracle told him "what [he] needed to hear." In other words, Morpheus expected the Oracle to tell Neo something that was likely not the truth. Trinity is with them when Morpheus tells him this and doesn't seem surprised to hear it. Neo has also told Trinity that the Oracle told him he was not The One.

Both Morpheus and Trinity are convinced Neo is The One and have no problem accepting that the Oracle might not have told him the truth.

Why are they so sure the Oracle told them the truth and told Neo what he needed to hear instead of it being the other way around?

At the time, Neo wasn't (yet) The One. As the Oracle says, "being the One is like, like being in love." She goes on to explain it, but she doesn't tell him everything.

She tells him he's got the gift, but he's waiting on something.

When he finds what he's waiting on (self-confidence and allowing himself to believe, love, and be loved) he becomes The One.

So what the Oracle says is true...from a certain point of view.

It's worth pointing out that the Oracle has never been seen to lie. So if she told Morpheus, "You will find the One," there will be no doubt in Morpheus' mind. When the Oracle is pinned down into giving a simple answer, with no layers of meaning wrapped around it and no room for ambiguity, she is 100% accurate.

Actually she never tells him he isn't the One. She just gets Neo to say it. She tells him just as Morpheus said what he needs to hear. Trinity doesn't seem shocked at the time because she doesn't think the Oracle told him he isn't the One. There's a scene later where she is surprised by that news. It goes back to the vase he breaks in the Oracle's kitchen. She tells him not to worry about it then he turns around trying to find it and breaks it. Then she points out that it will really mess with him when he thinks about whether he would have broken it if she hadn't said anything.

Oracle: You know what that means? It's Latin. Means 'Know thyself'. I'm going to let you in on a little secret. Being The One is just like being in love. No one can tell you you're in love, you just know it. Through and through. Balls to bones. Well, I better have a look at you. Open your mouth, say Ahhh.
Neo: Ahhh.
Oracle: Okay. Now I'm supposed to say, `Umm, that's interesting, but...' then you say...
Neo: But what?
Oracle: But you already know what I'm going to tell you.
Neo: I'm not The One.
Oracle: Sorry kiddo. You got the gift, but it looks like you're waiting for something.
Neo: What?
Oracle: Your next life maybe, who knows? That's the way these things go. What's funny?
Neo: Morpheus. He...he almost had me convinced.

The Oracle knows that you can't convince someone who doesn't believe in themselves that they are the One, so she sets Neo down the path that will make him the One.

Telling Neo that he was not the One as a lie may have served a specific purpose (e.g. lead him to develop).

Telling Morpheus that he would find The One when he would not served absolutely no purpose as far as the Oracle was concerned. As a note, I think that Oracle told him he would find The One, but NOT that Neo was The One.

Ditto Trinity. There is no plausible win/purpose for the Oracle to lie to her about loving The One.

I think there was some subtle word-play going on here. Neo had his own image of what "The One" was and meant, and he was not that person. So the Oracle told him the truth - he was not his imagiation of The One.

That he was Morpheus' idea of The One is not necessarily connected.

In much the same way that Jesus was not The Messiah that many of the Jews expected, but was The Messiah that had been prophecied.

Just like how the Oracle says to Neo "Don't worry about the vase" than he breaks it, shortly after she says "Whats really going to bake you noodle later on is, would you still have broken it if I hadn't said anything". Things wouldn't turn out the way they did if she told him the truth. If she didn't say anything about the vase Neo wouldn't have broken it but because she did things turned out they way they did, by her telling him that he was not the "One"that let Neo develop as a person to find his own answers rather than her telling him everything.

~Stefan Bozic

When she tells Neo, “you’re not the one. maybe in your next life”, she was telling the whole truth. He dies at the end and immediately comes back to life. It’s then he’s the One