I don't know if solfege is unavoidable (sounds like plague or flu!), but it can be highly useful. You can always use numbers instead...number the notes 1 to 8, or if you want to include all the possibilities, 1 to 12.
Two ways to start:
1. Melody playback: have the student close their eyes. Play a major triad and tell them what it is, then play a tune, no more than 4 notes long, based on the first three notes of the major scale. Play it twice, then have the student open their eyes and attempt to play the tune. I always have them play the triad before attempting the tune. If four notes is too long, just work on recognizing which note you started on (of doh, re, mi), and getting the second note -- or even just realizing whether it went up, down, or stayed the same. When the student is pretty confident on the melody using 3 notes, extend up to 4 or 5. As you do this, make sure you talk about intervals as well, and demonstrate how a minor and major 3rd sound different, for example.
2. For actual recognition of intervals, have the student sing a major triad up and down: doh mi so mi doh, then work on the intervals within the triad. The descending minor 3rd is the easiest to sing/recognize, so you could play descending minor 3rds and perfect 5ths for the student to identify. Then sing as follows: doh-mi major 3rd; mi-so minor 3rd; soh-mi minor 3rd; so-doh perfect 5th...that sort of thing.
2b. Extension: sing the scale, returning to doh before every other note. Doh-re major 2nd (slight pause); doh-mi major 3rd (pause); doh-fa perfect 4th, etc. When the student is comfortable with this, come down, covering minor intervals. (Doh'-ti minor 2nd, etc.).
2c. Extension: sing a 4-note chord up and down: doh mi so doh' so mi doh, then the intervals within as follows: doh-mi major 3rd; mi-so minor 3rd; so-doh' perfect 4th; doh'-mi minor 6th; so-doh perfect 5th. Then, stay on the same starting note, but call it mi (in other words, if you've been working in C, start on C but in the key of A flat major). Your new arpeggio will be mi-so-doh'-mi' - doh'- so-mi, and the intervals will be minor 3rd, perfect 4th, major 3rd going up and major 6th (C' to E flat) and minor 6th (A flat to C) descending. Then repeat, using C as so, and working in F major -- giving perfect 4th, major 3rd, minor ascending; perfect 5th and major 6th descending.
2d. Extension: do the same as 2c, but using dominant 7th chords (so ti re fa re ti so) , then diminished 7ths.
Any of the above can be done with numbers instead of solfege, but I think it might be confusing singing "8-3 minor 6th"...