If I may jump in. I believe your question is how much of the song do you play. I think you will find that with simple tunes all the verses will contain the same melody, different lyrics, but, same melody. If you know the melody to one verse you know the melody for the entire song. Yes the chorus may change. The point of the exercise is to get you playing some
melodies. So, to your question, one verse or at least enough to where everyone can recognize what song you are playing.
Somewhere over the rainbow comes to mind. First 7 notes of this should give everyone enough to know the song you are playing, but, go on and finish one verse.
Art makes a good point of leaving room for the tune to breath. The vocalist will be pausing, your tune should also pause, that e-book I listed points to three close notes then a leap of at least a 3rd. People like to hear four note phrases. A string of notes becomes noise quickly, however, if that string is broken into phrases we like to listen to that.
An example of improvising a melody. http://www.youtube.com/watch?v=r0iZ1j00wSU
Copy down the hints that appear on screen.