I would say you're on the right track. You've already noticed a pattern of jumping from the 5th to the 3rd to the 6th. That's a pattern of two intervals - a minor third down and then a major 4th up. If I were looking for melodic signatures I would look at intervals and the rhythms they're played or sung in. What are the relative lengths of the 5th, 3rd and 6th? Does Ben Folds repeat the rhythmic relationship of those notes? Rhythm and pitch are the two main components of a melody.
A pattern might also have a typical location where it shows up. James Taylor has a pattern in the second bar of a four bar phrase, where the 2nd note of the tonic scale is sounding on the first beat and by the 4th beat he's moved by a major second interval to the 3rd. The rhythm in between varies somewhat.
Pick another composer, style or period and listen until you hear a repeated pattern and then study it in the same way.
Next, you might work outside the linear realm of melody and start including the harmonic dimension. I can often tell Beethoven or Brahms after just a few bars based on their characteristic harmonies (I realize that's not pop).
One way of thinking about harmony is by looking at the chord progressions. Patterns occur in chord progressions just like they do with intervals. Certain chord progressions might also be used repeatedly with the same rhythms or in the same parts of pieces.
Finally, what instruments are favored in different ways? You might find that some instruments are typically given one pattern to play and other instruments another pattern. Or maybe the pattern is repeated by different instruments. Either situation could be part of the musical signature. Instruments can take roles, too. Does one instrument play the melody more often than any other? Which instruments typically back or embellish the vocals? Music from the same periods and genres often share the same instrumentation.
