Stratospheric aerosol injection (SAI) can provide global cooling by adding aerosols to the lower stratosphere, and thus is considered as a possible supplement to emission reduction. Previous studies have shown that injecting aerosols at different latitude(s) and season(s) can lead to differences in regional surface climate, and there are at least three independent degrees of freedom (DOF) that can be used to simultaneously manage three different climate goals. To understand the fundamental limits of how well SAI might compensate for anthropogenic climate change, we need to know the possible surface climate responses to SAI by evaluating the SAI design space. This research work quantifies the number of meaningfully-independent DOFs of the SAI design space. This number of meaningfully-independent DOF depends on both the climate metrics that we care about and the amount of cooling. From the available simulation data of different SAI strategies, we observe that between surface air temperature and precipitation, surface air temperature dominates the change of surface climate. The number of injection choices that produce detectably different surface temperature is more than the number of injection choices that produce detectably different precipitation. At low levels of cooling, only a small set of injection choices yield detectably different surface climate responses. As more cooling is needed, more injection choices produce detectably different surface climate. For a cooling level of 1-2C, we find that there are likely between 6 and 12 DOFs. This reveals new opportunities for exploring alternate SAI designs with different distributions of climate impacts and evaluating the underlying trade-offs between different climate goals.