What can drive the higgs mass in mSUGRA up? As a test to familiarize myself with the program softsusy, I generated a spectrum for the following (already excluded, but once considered) mSUGRA point: $m_0 = 170\,\mathrm{GeV}$, $m_{1/2} = 190\,\mathrm{GeV}$, $\tan\beta = 3$, $\mathop{\mathrm{sign}}{\mu} = +1$ and $A_0 = 0$. The result is displayed in the picture below. While the mass hierarchy looks OK, I'm worried that the lightest scalar Higgs is around 100 GeV.
Now, as we all know ;-), the Higgs lives around 125 GeV. Also, I understand that in leading order the lightest higgs should be around the Z mass, and that it gets a large correction from the top quark that pushes it upwards. What I'm wondering is now:


*

*Is is really a feature of mSUGRA that the higgs is so light? Why have people been studying points like this, when this would have been ruled out by LEP? The $h_0$ should be sufficently similar to a "SM" Higgs, right?

*If this is not a genuine feature, have I made a mistake? Is there a way to persuade softsusy to add higher order corrections that drive the mass up? (I'm already using the "Include 2-loop scalar mass squared/trilinear RGEs" option.)

 A: I think it simply means that you didn't choose the point in the parameter space that gives the Higgs mass you want. Try and increase $\tan \beta$ and/or $m_0, m_{1/2}, A$ and see how this affects the mass of the lightest Higgs. And yes, this little $h_0$ is a SM-like Higgs. 
To answer the question in your title, the mass of the lightest Higgs in the MSSM is given by $$ m_h^2 \approx M_z^2 cos^22\beta + Loops, $$ and $$ Loops = \frac{3m_t^4}{2\pi^2 v^2} ln(\frac{m_{\tilde{t}_1}m_{\tilde{t}_1} }{m_t^1}) + Mixing $$ where the mixing term depends on $A_t$ and other parameters. 
So if you want to increase the mass, you have to increase $\tan \beta$ as well as the mSUGRA input parameters, because they determine the average mass of the stop (and the mixing)
A: This paper introduced the so-called $m_h^\mathrm{max}$ scenario which maximizes the highest possible value for $m_h$ for each value of $\tan \beta$ (and for a fixed value of $M_\mathrm{SUSY}$, note that the top mass has decreased a bit since the paper was written).
Looking at Figure 4, it looks (at least with their choices of other parameters), that the mass of the light CP even Higgs boson does not go beyond 132 GeV.
The fact that a particle consistent with the Standard Model Higgs boson was discovered below 132 GeV allows for the possibility that this could be the lightest CP-even Higgs boson predicted by Supersymmetric theories. 
The MSSM would be in serious trouble if such a particle had been discovered at a mass much above 132 GeV.
