Ideally the desktop app would be a GUI which was easy to use, providing the basic features of Word or LyX or a markdown editor such as basic formatting, bullets, headings at various levels. Ideally some kind of diagram editor as well.
Ideally it'd save in a format which was reasonably amenable to being saved in Git and being merged from various branches, and which wouldn't be too verbose if you're looking through a PR on GitHub etc. So a text format, with no "# Created on <date>" fields. I have to say LyX solves most of these points so that's what I use at the moment.