For someone that doesn't have a concept of harmonization, you have to teach them
a harmony part that works and have them sing it as if it were the actual melody of the song. If you do this with enough songs, they may get the idea.

In the meantime, singing harmony is a cultural thing. You usually grow up with it.
It can be taught as an adjunct to formal music theory as part of a sight-singing course.

For a guitarist, one possibility is that the singer can learn to sing a part of the chord
played on the guitar that harmonized with the melody.

The best way to learn harmony is to do a trial and error approach where the learner
composes a harmony part and sees where it works and where it doesn't. This is the
best way to learn aside from formal music theory.

The most simple parts are best to start in which there is one note held throughout most of the harmony parts. For example, the fifth note of the chords that are being harmonized is held and will harmonize with most moving melodic lines. Eventually,
the learner will begin to hear the third note of the chord. The basic harmonic tones
for harmonizing are the fifth note and the third. The third becomes the sixth inverted.
Simplicity in harmony requires hanging on to one note as much as possible.

Frank Hamilton