A melody can up or down, or some combination of the two, and it can move by steps or arpeggiated, as you like it. It can even stay still for a moment, it can create and resolve tension. It has a logic: there is a phrase, and then another phrase that repeats it or answers it. It all makes logical sense. All this is something anyone knows. A melody is not a random collection of notes, but a series of phrases in logical succession.
But what I think a melody needs is a lilt, a hitch, or a catch; a knot or point of some kind. Something that makes it not just a logical succession of notes, but some other little element that catches on the musical brain somehow.
Imagine that a hand moves over smooth silk. Any rough, uneven spot on the hand will catch on the smooth surface, so there are not two smooth surfaces interacting but one (partially) rough instrument testing itself on another with sensitive fibers.