I believe if you're doing that to get at something you use the DDOS to hide your true intentions. e.g. you leave the DDOS at the level to slow the site way down so that the admins are paying attention to that while you go after what you need. It's like blowing up the building across the street during a bank robbery, you distract the people looking for you so that you can get away.
If you can get a crash, you can possibly find a weakness. If a system relies on Github to work properly, maybe you can find vulnerability in case Github is down. Maybe a MITM (to your target, not Github itself) is easier to perform when Github is slowed down and takes some minutes to respond to requests.
There are various possibilities, nevertheless, bringing down Github is surely a juicy objective.