I've now had a few days to mull this over, as well as read the thoughts of others. Here's what I think about the change:
First, I am extremely disappointed at how slapdash this feels. It seems like at no point did anyone ask any of the important design questions beyond "what can we do to mix up ARAM?" No one seemed to ask things like, "what problem are we actually trying to solve?" or "how many options do we have to fix this?" or, "what would be the consequences of making these changes beyond solving our problem?" This is why, I feel, we ended up with a mechanic lifted completely unaltered from a goofy game mode and placed into another game mode that, while not as competitive as ranked SR, was still taken quite seriously by many of its players.
This, I think is the first major problem: this new mechanic was the fundamental mechanic of the Poro King mode. When you ported it wholesale into ARAM, you then made it the fundamental mechanic of ARAM as well. ARAM was always defined by making strange team compositions work together using the mechanics of the champions, but with the introduction of the snowball it's become a snowball fight. Everything now is about dodging around the snowballs. Maybe it's just me, but a mode where the whole game revolves around a single universally available mechanic feels far more like a joke mode than the old ARAM ever did.
The second major problem is that this mechanic obviously wasn't play tested. Even if we discount its horrendous imbalance, the fact that you have to come back and do damage control with it shows me that no one fully thought about the consequences of the snowball's implementation. Now that we've all seen it in action, we all know how it changes the game. It does unnecessary damage at rate higher than Ignite, it has double the range of a Nidalee spear, it has a cooldown shorter than some regular champion abilities, it isn't blocked by spellshields and Windwall, and it can be used on minion waves just as easy as it can champions. It is almost pure reward for no risk - the only danger is when you actually take the dash. This is not balanced, because it was built for a temporary game mode and was never meant to be a part of anything else.
The third major issue is that it's still unclear on what problem this mechanic was supposed to fix. Was it supposed to make all those 40% win rate champions viable? Because it didn't. In Mandaari's excellent analysis, he shows that the problem with these champions is almost universally not their lack of engage, but rather the fact that the geography and terrain of HA isn't conducive to their strengths. I think the most obvious example is Rek'sai, who is all about multi-path mobility and jungle control. But, HA has only two directions and no jungle, so by no fault of her own Rek'sai is rendered bottom tier. Giving her the option of more engage isn't really going to change that, because this mechanic isn't just beneficial to her.
If the snowball was supposed to help out the bottom 5-10% win rate champions, why does it benefit 90%+ of the champion pool? The only champions that really don't benefit in any way are those with nothing but poke - Lux, Xerath, and Ziggs etc., and most ADCs. Every other champion would be five kinds of foolish not to take this ability because it's just too good. In effect, this ability does more to make good champions better (Alistar, Amumu, Galio, etc.) than it does to make bad champions good (Evelynn, Tryndamere, Udyr, etc.).
Bringing up poke champions, though, will inevitably lead to talks of ARAM-only accounts stacked with the high win rate champions. But this mechanic doesn't solve that problem either. Sure, it feels like karmic vengeance now, while those people with a large number of unlocked champions can drop Tibbers on a poke-heavy team by engaging from a screen and a half away, but there is nothing stopping those people those people who gamed the random champion mechanic from making a new account and unlocking only champions that are now the best with the snowball. At best, this change has only given people a temporary reprieve from ARAM smurfs. I would not be surprised in the slightest if we see a new wave of ARAM-tailored accounts crop up within a few weeks should this mechanic stay as it is.
And to be honest (and I'm trying not to belabour this point, but it is important), the lack of engage was not really the issue with the low win rate champions. The problem with a team of bruisers vs. a poke comp wasn't any problem with the champions or summoner spells or items (especially after the addition of Righteous Glory), but rather with the patience of the players. ARAM used to have this really neat 'race against time' dynamic when a team of tanks met a team of poke. The poke team needed to win before about the 15-17 minute mark before the tanks got too strong, and the tanks needed to survive until then. The problem is that at lower MMRs, players don't have the patience or game sense to play safe and smart until they were strong enough to take a team fight. Now, the tanks just start diving at level 6 because they can just bypass the poke.
That's not to say there weren't difficulties. Mandaari has already pointed out the problems with waveclear and tower siege with melee champions, and there is still the issue that certain champions simply won't be good due to the map geography. But these issues could be solved in much more nuanced and appropriate ways. The suggestions that Mandaari made are all good, and all address specific problems, rather than the vague feeling that 'melee champions are bad'. But the addition of this mark really did feel like trying to use a snow shovel to eat your ice cream: a spoon would have done much better, and it wouldn't have been so intrusive and unbalanced.
So, what would I do with the snowball? Personally, I think it should be at least temporarily removed. If it's something many others think is worth having, then it needs a hefty rework and a proper testing period. That said, I'd still rather the problem be fixed by tweaking mechanics already in place rather than adding an entirely new one. And most of all, any solution, should have a clear purpose. If the goal really is to make champions like Udyr viable, then the change should be beneficial specifically to them, and shouldn't be abuse-able by champions who didn't need the help.
- 3 of the most op champions in aram, and guess what... they are melee! They have gap closers, and the ability to do a lot of burst damage once they enter a fight. They barely have any poke compared to some other champions like 
or 
yet they still enter a game destroying the other team. However, I have played plenty of games against an Akali, Yi, Katarina and won them, sometimes even on a melee champion! And there are other times I had one or more of these OP champions on my team and we still lose.
and guess what, he's melee too! However, I have Ice Blast and Absolute Zero to slow the enemy down, as well as Bloodboil to speed myself up! I do not need a Mark/Dash to get into the fight if I play right and time my abilities at the proper time.
with his grab 
if he lands the knockup 
whither 
. it does little for those who already did little 
combo makes him supper sticky and hard to get off your butt, and the 
another 