Does the Howling Abyss map (ARAM map) use an MMR rating system?

Solution 1:

Yes.

Source: Riot Pwyff

I know we have separate MMRs per map. I believe HA will have a new hidden MMR specifically for it but we get that ARAM is less about precise skill matchmaking and more about SUPER RANDOM CHAOS. So the focus is on light balance and more on getting you matches in the queue faster.

Source: Super Nomegeta

It'll have its own hidden normal Elo, but be based on your SR normal Elo initially.