Reharmonisation of a melody is a very attractive way for musical development and is constantly used in film music to shape a theme according to the mood that is needed. The possibilities are almost endless and you could reharmonize by just adding a few extensions to the chords up to completely changing the harmonic progression or even tonic.
To reharmonize, analyse your theme for important melodicy notes. Let’s say you have a melody note of C which originally was the root note of your tonic. When reharmonizing, you have many different options. That C could become the fifth of an F major chord, third of Am or Ab, maybe the maj7 of a Db, the 9 of a Bb, #11 of an F# etc. Some of these solutions will work better while others won’t depending on the chords and melody notes and chords that surround that C. Apply the same principle for every of your important melody notes to find a new chord progression. In order to not get lost completely, you could also say that every 4th chord (depending on your form) of your original melody will be as in the original, which will ensure that you most likely stay in the same tonic and have the same dominants at the end of your melodic arcs.
It takes a bit of training and experience to be able to do that without struggling but once you get the hang of this, you’ll be able to emotionally shape almost any melodic idea. Of course, you need to make sure that the new chord progression still has musical cohesion, so random reharmonization of individual notes will most likely not work. Also, some melodies limit the possibilities for reharmonisation quite heavily. This applies particularly for melodies that rely a lot on arpeggiated chords.