Take a look at this interesting video which shows different methods to come up with songs, melodies and beats.
As seen from the video, writing can start with either a singer singing a melody, or a simple synth or a chord progression. This can easily transform into a lengthy jam session. It is important that everything is recorded in order to reference it back later. Sessions that may not have sounded too good at the time can often sound awesome when you listen to them another day with a fresh mind. I would say that you don’t even need lyrics for these type of Jam sessions, since they are simply ideas. As Solange mentions, these ideas can them be taken elsewhere to develop into full tracks and add vocals. I love how she plays those simple keys and chords at the beginning and then transforms it into a song.
Personally I like to start of with chords, since chords hold so much information about a song. e.g. Keys, Bass and melody. Starting of with just a synth or a bass can make it hard to tell which direction the key is going, however everybody has their different methods.
Song writing in Japan
Playing with Japanese here in Japan, I find that Japanese prefer more structure then random Jams. Jams do work here however the ideas will be quickly extracted out of a jam session in Japan and band members may cut the jam short to focus on the structure in the early stage. The foreigners (Gaijin) may be happy to jam all night for that particular session. Maybe leave the structured session for another day. Sometime as a musician, you may not even be in the right frame of mind for a serious structured session. I have been playing in studios doing back to back serious structured sessions and I had been dying to just jam or freestyle, but before you know it the time is up and staff come in trying to kick you out so the next group can set up.