A Legitimate Problem: ARAM Stacking
See bold for TL;DRs
###What is ARAM stacking?
ARAM stacking is when people level/make accounts strictly with the purpose of ARAM play. They usually only buy champions very strong on ARAMs and end up dominating with "stacked" odds in their favor. It's a problem and often subtracts greatly from the "random" aspect of ARAM.
###But surely there aren't people who do that, right?
See here (Name withheld because no calling people out)
At first, it looks pretty standard...but look closer. Notice how every match is an ARAM. Further more, all of these are considered "strong" ARAM champions. More so, look at how some of these are obtained twice in the span of just a few matches.
This level of coincidence is beyond simple luck. This is stacking the odds.
###Yeah but even with stacking the odds he can't play these guys all the time!
Let's provide an example. Let's say I'm a summoner with these champions:

Ten "strong" ARAM champions with poke, sustain, etc. (Just roll with me here)
Now let's propose a "worst case scenario" free week.

Even with this, the chance of getting a champion that's ARAM-strong is 50/50%. You have a 50% chance with these odds to draw a power pick. Keep in mind, I set up this to be the one of the worst free weeks possible. You will likely encounter free weeks where you own several of the champions (Further skewing the odds) or you'll own champions suited to ARAM beyond these ten.
###Even so they have the reroll disadvantage becau-
The most common argument I see on this but it STILL doesn't negate the problem. Basically what this tells me is "If they botch two rerolls they'll be stuck with a champion they hate". This is not a fix to the problem. At best, this will discourage people from doing this. At worst, this won't do anything.
###Whatever, ARAMs aren't a serious mode anyways.
And that makes it ok? Imagine if I hacked my client (extreme example but bear with me) so that I only got matched up with professional players who carried me every game while the enemy team was always bronze. Sounds unfair right?
It's 5v5 blind pick though, it's totally ok. It's Dominion, it's ok. Etc etc.
The moment you let this argument prevail, you are basically opening the floodgates. At what point do we draw the line? Will it get to the point were any mode that's not 5v5 ranked is just considered "Meh w/e it's not serious" and we treat it as a-ok?
###So how would we fix this?
Tons of solutions. For example...
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Make ARAM -actually- an ARAM. All champions enabled. Riot doesn't lose money because it's still random, ARAM is actually random, nothing of value is lost. Heck, more people might even play ARAM to test out champions they don't have.
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Reward people who have more/all champions. If you own everyone or 100+, maybe every game you get one free reroll. Maybe you make it 3 reroll stock instead of 2. Whatever works. Just don't let these guys keep having little-to-no punishment.
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Force a champion minimum. While this is probably the least effective, it might do SOMETHING toward stifling the ARAM fixing. Even so some action is better than NO action.
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Add a stacking "debuff" to champions. For every game where you get gangplank, for example, your chance of drawing gangplank is decreased. This lasts 30-ish games or so. Sure, doesn't solve the problem but it's a step towards it.
###General Conclusion
Nobody likes playing a fixed game. Nobody likes having to face a team of:

while your team is basically:

While Riot has said before "Oh we don't care people suffer enough", they are wrong in this regard. ARAM stacking is a clear, obvious crime and nothing is being done to fix it. So please, if you can take one thing from this entire argument?
"Never lost a fair game...or played one."