This particular algorithm (Stevens & Olsen 2004) uses a recursive quadrant randomization based on tessellation of the data. The "inclusion probability" raster is used to normalize sampling intensity, functioning as sample weights. If your data is spatially random you can provide a uniform weights raster (all 1).
The issue here is that, theoretically, the inclusion probabilities should represent an intensity function. Not exactly sure why the authors of the tool did not include this, seemingly obvious, component to the model as an option. Without coding it in Python/NumPy I do not think that you can produce a spatial intensity function in ArcGIS. The spatial intensity is akin to a Kernel Density but, represents the expected frequency/density of the observed point process.
Ideally, the probability weights would be based on a model of your process. If you have the data to support it, a method such as probability kriging would produce a surface that could be used as an inclusion probability raster.
It is not entirely valid but, I imagine you could derive a Kernal Density estimate, scaling it to [0-1] using ( d / max(d) ) to use as the probability weights however, the results will be highly dependent on the parameters used for the density estimate (ie., bandwidth). The trick here is to decide on what order of spatial variation you want to target. This will be entirely dependent on the bandwidth used for the KDE. A small distance bandwidth will converge on second-order (local) spatial variation whereas large distances will represent a first-order (global) spatial process and notably smooth the data. A KDE can be created in ArcGIS (with a Spatial Analyst license) using the the "ArcToolbox > Spatial Analyst toolbox > Density > Kernel Density".
Stevens, D.L., & A.R. Olsen (2004). Spatially balanced sampling of
natural resources. Journal of the American Statistical Association