I think such bands as Brethren, is fine example where it hardly in any way qualifies as "noise". Of course its rarely said to be noise, but especially true for later works, he creates songs with very distinctive rock music template of verse/chorus system. Even if its generated out of noise loops.
I don't consider melody or specific rhythm to be essential, but some moment what will make long lasting impact on your memory is always good thing. One of my absolute favorite noise pieces is this:
https://www.youtube.com/watch?v=hhAPBQugBtEOf course it has rhythm patterns as well as lots of other texture, but there is some unmistaken personality in track, that if someone puts it on speakers, I know exactly where it is from. This is not same with a lot of other noise tracks. Tracks which may be great, but without proper hooks or distinctive sounds that have long lasting effect. It can be ok to just deliver the goods in moment of listening, but I personally also value highly ability to maintain energy beyond moment when you push stop.
Another track I must have mentioned several times before, is Emil Beaulieau "Schimpfluch" (track #7) on Memories CD. It has repeating pattern ripped off from Schimpfluch works, that is repeated several times during song. Very unique, sort of "robotic" changes of intervals of electronic tone. Even if one could give credit to Schimpfluch crew rather than Emil, still that tracks is great piece of noise, but you can't mistake it to any other noise piece made in this world.