Answer: Writing a counter melody is kind of a big deal. And as in the wild west, there are no strict rules.

It's important to remember that the countermelody is not just a harmony part for the first melody. It's making up the second voice in a conversation. Frequently the countermelody might become more active at times when the first melody is on long notes or resting - so that the two seem to be talking back and forth.

The second melody could either contrast or resemble the first one. For making a family resemblance that preserves independence, the devices of tonal inversion and tonal transposition are useful. By "tonal inversion" is meant making the second melody zig where the first one zags - if the first melody goes up a third, the other could go down a third. Being "tonal," or within the key, means that an upward third in the first one might end up being a downward minor third in the other, but you still have some resemblance between the two. Tonal transposition means moving parts of the first melody up a step or two, but again using just notes of the key. The countermelody might be expected to begin after the primary one sets the tone.

If you're going for contrast, a contrast in rhythmic movement will help to distinguish the countermelody. Rhythms can be augmented (longer versions of note values found in the first melody) or diminished (shorter). A vivacious melody might be countered by a more solemn and slow one. A good example of the contrasting counter melody is found in Bach's famous setting of the chorale melody "Wachet auf, ruft uns die stimme" in his Cantata 140. In this excerpt displayed in Songworks we'll let a trumpet take the part of the tenor singer, and a flute (a pan flute in the synthesizer) will do the counter melody. The both are underpinned by the bass, here played on single notes of a piano. Notice that the chorale melody is slow and stately, while the counter melody is more lively and flowing. But both follow the same harmony - that's what binds them together:

To my great regret I have not been able to locate a recipe that would enable me to write counter melodies that could compete with the above from Bach. Or anything else from Bach. But here is a general idea that might help. Leaving aside all considerations of counterpoint for the time being, just think of the harmony. Assuming that your first tune has a good solid chord progression behind it, make a melody that contrasts in character but works with the same harmony. Then try putting the two melodies together. Often even random melodies will work surprisingly well together if they follow the same harmony.