I also needed to know how to rename bands in an open source environment so spent a whole day looking for answers.
There is no way to name bands in QGIS while merging.
But it can be done after the file is created, by editing their .aux.xml file. It works for both .tif and .img files, as far as I've researched.
The solution is to include after each
<PAMRasterBand band="2">, etc, a subelement
<Description>YourBandName</Description>. I haven't tested it for VRT datasets, but the link at the end suggests that it should be the same procedure.
In ArcMap, you will be able to see the new names in the Properties > Symbology window, when setting your colour composite, also in Table of Contents, under your image layer.
In QGIS, I suspect there is a bug, and this will not work. However, you can still label your bands in a more indirect way by also including after the element the subelement:
This will show up in the Properties > Metadata tab, for each individual band.
Example image attached, only first band was named for demonstration. Checked on ArcMap on Windows and QGIS in Ubuntu.