Writing your own banjo songs is a great way to find your own voice in your instrument. It can also help your playing quite a bit as it can get you out of your routine of playing the same things. It's fun too!
I often can come with original songs on the spot, but can never remember them. I now try to record these tunes in order for me to be able to remember them and to come back to them later and work on them some more or play them again. You don’t need a fancy recording system. I just use the voice recorder on my iPhone.
This goes along with tip #1. If you aren’t recording it audibly, then try writing it down in tab so there is a record of it. I prefer to record it first, and then come back to it and transcribe (write down in tab or musical notation) what I played.
If playing tenor, plectrum, or 6 string banjo, you might then play chord voicings that feature that melody note on the top (1st) string. If playing 5-string, you can play a chord shape with that melody note on the 1st string, but use a 3 finger roll or clawhammer strum to play the chord.
This is similar to tip #3, only in reverse. Come up with a chord progression ex. 1 bar of G, 1 bar of C, 1 bar of Am, and one bar of D. Record yourself playing this progression. Then play the recording back and come up with a melody.
The other day I was in a rehearsal and another group member played an "original" tune. It sounded kind of familiar but I couldn't place it. What this person had done is to take a standard tune that is in a major key and changed the melody so it was in a minor key. He then put a different rhythm to it as well. Now this may not stand up in court as an original tune, but it does give you something new to play. This is just one example of how to do this, but using a pre-composed tune as a foundation and altering it or "borrowing" some of it has been a very common way of composing new tunes throughout the history of music.
Please feel free to ask any questions or post your ideas in the Comments section below. Have fun!