This is a very ambiguous question. When you start talking about particle theory, there is an entire collection of properties that you could call the mass of the particle. For quark, there is the constituent mass, current mass, and the running mass. Each one can be estimated given some assumptions.
Constituent mass is the most straight forward one. It's the fraction of hadron's mass for which a given valence quark is responsible. Unfortunately, because this includes all of the sea contributions, this is going to be entirely dependent on the properties of the particle and not on the quark itself.
Then there is the current mass. That's the thing that goes into Lagrangian of the theory. That's the mass that would be zero if it weren't for the Higgs Mechanism, and I suspect you could make some estimates based on that, but I'm not really an expert on Higgs, so I don't know what the state of the art there.
Finally, there is current mass. If quark did not interact with anything, it'd have the propagator \frac{i}{\gamma_{\mu}p^{\mu}} same as any other elementary fermion. Here m is the same current mass as above. But it does interact, so instead of constant mass, you get running mass m(p²). This can be computed from Gap Equation for certain values of p² under a whole list of assumptions and with a bit of phenomenology mixed in.
And it's worth noting that for most particles, there is one more important case, which is what you'd most commonly call mass of a particle. It's the solution of m²(p²) = p². In other words, requirement that momentum is on the mass shell of the particle. Such a solution exist for any free-propagating particle, but since quarks are confined, there is no p² at which this is true.
