There are a couple of different options for changing the colour of particles in a particle system with respect to their lifetime.
The first is Gradient, that let’s you set a gradient of colours that the particles change through over time. You can add a number of colours to the gradient and position them to get a desired blending, and so the colours occur at particular points in the particles’ lifetime.
The second is Random Between Two Gradients, which is similar, but let’s you set two gradients with the colour of the particles being a random blend of them over their lifetime.
These are different options to the other …Over Lifetime settings, as it doesn’t offer using a curve or a constant value (or random values between them). Ultimately though, a gradient is more convenient to use to set up a change in colour over time (and is a sort of curve, though not visualised as one).
You can see the results of my fiddling around with the setting below: