First, you both will need patience. Learning to distinguish sounds from one another is not a simple process. Imagine if you were color blind and had to reproduce the color blue after only seeing it flash in front of you for a moment.
Of course, the obvious difference here is that there is little to be done for color-blind people, and hearing notes / music is basically making the blind, see.
Pursuing this endeavor facilitates Occam's Razor: you need a piano / keyboard. (I'm going to paraphrase several year's worth of childhood aural development into a few bullets below). The bullets are also accounting for the fact that you would be working with an adult, otherwise, I'd leave the animal sounds in:
1.) Start by playing one or two notes at the extremes of the piano. Establish definitions for "high" and "low" sounds (not related to volume).
2.) Ask which sound is higher/lower than the other. Repeat as many times as it is interesting - you can make games here, even turn it into a drinking game if you'd like.
3.) Very gradually, over time (a few weeks of regular practice), slowly bring these extremes together such that after several weeks, the person can play this game using a variety of simple intervals, chief among them major / minor seconds. If you have a string instrument (violin, viola, etc) you can play micro tones to get even smaller intervals.
4.) In parallel with these games, also employ simple melodies (ABC's/Twinkle, Happy B-day, Wedding March, their favorite music, etc) and have them show you the relative pitch of the melody with their hand - the higher the melody, the higher the hand, the lower the melody, the lower the hand. Once they become more proficient, you can teach them the Kodaly hand signs.
4a.) You can extend this by playing various scales in both directions, as well as jumps of various sizes. You can create melodies, improvise them, or have them create a melody and show the shapes.
4b.) Have them teach you these exercises and how to listen for high / low sounds.
5.) Once they have a firm grasp on high/low sounds, introduce matching pitch. Have them be able to tell the difference between someone matching pitch and singing a different pitch.
6.) Have them try to match pitch (visualization: tell them to fit their voice "inside" the sound) as well as try and sing a different pitch. Sometimes if they are a little off, you can coerce them to the desired pitch by meeting them where they are and sliding them up chromatically to the desired pitch.
6a.) Side note, make sure they can physically sing the notes you're asking them to. Children often have a difficult time matching pitch not because they can't hear it, but their voices aren't physically developed enough to do it.
6b.) There's more but I need to stop myself before I end up writing out a whole curriculum.
7.) Additional random games to further develop aural skills:
- 1 note game: play a note, have them play/sing. Each time they get it right, add a note to the sequence and make it longer, playing the same sequence.
- Play triads and omit one note; have them sing the missing note.
- Play a note, have them sing a particular interval above/below the note; move it chromatically/diatonically up/down but have them maintain that interval throughout.
- Teach them to sight-sing with solfege.
8.) There's always a million more things one could write about this stuff, but hopefully this will give you and your friend/relative a starting point.