I wanted to provide some information after working with ESRI technical support on this issue.
Firstly, this tool does not work with hyperspectral data. There is an open bug regarding this issue (BUG-000143872).
I did find out some critical information about how the tool works for non-hyperspectral rasters. To update the band name information, the "Name" field in the Key Properties section should contain the current band name (e.g. Band_1, Layer_1, etc). The help documentation was a bit ambiguous.
Set Raster Properties to update band names
I also found the Get Raster Properties tool to be helpful to gather my hyperspectral band information. I used the tool on an original raster with the band properties still intact:
Get Raster Properties to collect band names
Using the wavelength function and leaving the band selection blank, I was able to run the tool and view details. The output printed a list of all the band names and the wavelength ranges. I copied this text and brought it into Excel where I was able to break apart the strings and format all the information properly. This made it easy to build a python script to update the Set Raster Properties tool.
Set Raster Properties python example
Ultimately, this did not work for my data, but hopefully the information is helpful for other users.