What is the Oracle's purpose, specifically regarding her "to unbalance it" comment?

The Machines have arrived at a balance in their effort to control humanity, but it is a fundamentally unstable balance. Therefore they must be on the lookout for a way to achieve a better balance. But before they could implement a new balance, the old balance must be perturbed. This is where the Oracle comes in. By understanding humans better, she can arrive at ways to creatively introduce a little chaos into the system, forcing it to evolve, hopefully towards a more favorable equilibrium (which has happened by the end of the third film).

[here I could make an analogy to an optimization technique called 'Simulated Annealing', but I think I'll leave that to the reader]

Oracle is a different form of control. That is her purpose.

Matrix is perfectly balanced most of the time. Most humans accept the simulation as reality. That is the purpose of the Architect: to create a simulation fitting for most, create an environment so the most crops survive and flourish. But, as in any population, there are variations.

So from time to time, in unpredictable intervals, the simulation starts to become unbalanced for small part of the population. Some humans refuse the simulation. They start to doubt the reality. Some of them even get to control parts of the Matrix, due to their genes or something machines cannot plan for.

Imagine a door which is 190 cm tall and 180 cm wide. Most people can pass through such door without bending. The expected average height of a healthy population should be 163 cm for women and 176.5 cm for men - as defined by the WHO growth reference standards. 99.5% population grows shorter than 190 cm door and thinner than 180 cm. Still, some few grow taller. What are you going to do with them? Are they going to have to bend every time? Or are you going to make a separate door for them? What if it is not a door but a tunnel? Would you keep forcing them to pass through a thing which is not right for them or will you deal with that in a different way?

The Oracle is there to deal with those who do not fit. She is there to give them illusion of choice. Because you have two different populations in Matrix. 99.5% which fit the mold and 0.5% which don't. The equation is perfectly balanced for 99.5%. The Oracle is there to corral, to move the remaining 0.5% in a way that is manageable and doesn't interfere with the remaining 99.5%. Let's call those Redpills. She is unbalancing part of the equation for small part of the population so they get the choice. There is literally one set of rules for 99.5% and a set of different rules for 0.5%.

But occasionally, one in 10 thousand Redpills or so is different. He is not your regular Redpill. He is The One. And every The One is unique. So there is very special protocol on how to deal with it. The One breaks all the rules. The One can't be balanced like you would balance regular Redpills. No, for that, Oracle first needs to find him or her then needs to scan and analyze him and create specific protocol to deal with him.

Create a loved one they would care about by manipulating one of the other Redpills. Or don't. Use The One's anger, love, courage, humility, greed... whatever the leverage you find during your scan. It is different lever for every The One. Because The One is an anomaly you cannot predict. He is the Black Swan, an event you cannot plan for but has dire consequences once he shows up. Like how you know that if you have 10.000 trains operating 24/7 365 days in a year, some of them will break down and some of them will have a huge train accident. But you don't know which one and how. Else you would prevent it. So there are protocols for regular operation, protocols for when the part breaks down. And then there are protocols for full blown train wreck. You try your best for train wreck not to happen, but still have protocols for WHEN it happens, a year, 10 years or a hundred years from now.

The Oracle's comment about unbalancing the Architect's equations is intended to emphasize the contrast between her and the Architect: he can't understand or see past choices but she can, and he tries to balance his equations but she unbalances them. The Oracle similarly contrasts Neo -- the One -- with Smith:

He is you. Your opposite, your negative, the result of the equation trying to balance itself out.

_{The Matrix Revolutions transcript}

A perfectly balanced Matrix like the original Paradise Matrix will ultimately fail (it was rejected by the humans), and it's obvious that the latest version of the Matrix trying to balance itself between the growing power of the One (Neo) and the virus Smith is going to fail pretty badly, too. Hence the Oracle is trying to unbalance the equation by supporting Neo, and she also believes that by doing so she can end the war in a good way.

The Oracle's purpose is still to investigate the human psyche (no other program is better at it -- except perhaps Sati, who is beginning to understand love and is hinted at being the Oracle's successor). It is this function of hers that allowed her to develop the far more successful choice-based Matrix to replace the Matrix betas. Her purpose is also to traffic information as the Merovingian was originally intended to do but which she also inherited as a result of her innovation of the choice-based Matrix. The Oracle's "unbalancing" of the Architect's equations is not so much her purpose as a means to accomplish her purposes.