Option is on File, Save DEM, menu when a map has a DEM associated with it.
The Reinterpolate options will create and open a new DEM. If you want to save that DEM, activate its window and use the Save DEM option. This will effectively reproject the DEM or grid.
Reinterpolation uses a bilinear interpolation, which MICRODEM always uses to get an elevation not exactly on a grid node.
There is no penalty in MICRODEM for using lat/long DEMs, and all computations use them without problems. Lat/long DEMs are much better for seamless operations (USGS and NGA agree, and that's what is used for 3DEP/NED, SRTM, ASTER, MERIT, and DTED covering moderate to large areas).
If you are downsampling and want to consider a larger region rather than just the four points around your desired grid postings, you can Filter the DEM first.
If you reinterpolate to a finer resolution than your original DEM, you will get a smoother, generalized DEM, which will behave inside MICRODEM exactly the same as the original DEM. The reinterpolation will not be "better" than the original DEM and will be arguably worse (high and low points tend to be smoothed out); if your DEM is lead, nothing you can do with it can turn it into gold.
Before reinterpolating, you should be sure that you understand the actual resolution of the DEM, and not just the appearance on screen because MICRODEM defalts to showing the entire data set on screen without scrollbars.
Last revision 9/17/2018