Rav Elyashiv apparently utilized a distinctive melody when learning. Are there any extant recordings, either audio or video, of this melody?
I've found this video of R' Elyashiv learning at home.
There is also this much longer video (with better audio), but the first 8:15 of this one is dubbed over with music.
Although it's longer, watch the second one, or leave it playing in the background while you do other work/learn, because you're right, in the recording he does hum and chant his learning with a tune that is somewhat different from the standard chanting commonly heard in most Batei Midrash in which I've learned.
here is one from about 30 years ago.