A simple way to add a little harmony here and there when playing the melody is to find on the middle string a harmonising note - it's usually the adjacent fret plus one over and play it at the same time as the melody.
So.. if your 4th finger was on the melody string, then your index finger would hop over to the middle string and be diagonal to the melody string fret with a gap of 1 fret inbetween.It's a clumsy explanation but if you try it you'll know immediately if it's wrong or right. I think this note on the middle string is the 3rd above whatever you're playing on the melody. Or the 5th? I've never gotten technical about it.
What the other posts were getting at was substituting chords which can sound very interesting (as long as you know what you're doing! lol) One easy way to approach it might be to substitute a minor chord for a major chord occasionally - at the end of a section for eg.
Which reminds me, the friends I play with are now doing You Are My Sunshine. It was decided to play the verses in minor keys and keep the chorus in the major keys. Definitely a different slant on the song! But you know the lyrics are a little on the dark side...