So, after the potion capacity nerf it finally dawned upon me that one could, get this, buy a Bead and then - shock and awe - not build it into anything and just sell it when the slot/gold is needed.
So lets examine the efficiency of this.
A red pot heals 150hp and costs 35g. A bead costs 180g and heals 1hp per second. That brings the gold efficiency of the potion to 4.3hp/g and the efficiency of the Bead to 0.0056 hp/g/s. Solving for time, you need to hold the Bead for just short of 13 relevant minutes. So that doesn't look so good. However, the Bead sells for 126g. If we assume we do this, the time threshold goes down to just short of 4 minutes. And of course the longer it is held, the more value it accrues. If we let health potions set the value of health regen, the bead produces a value of 0.23g/s (14g/min).
Now, this isn't exactly the whole truth. The potions can be used up much faster, giving more bursty healing, which lets you use the extra health when it's most valuable.
It does look like potions are altogether a bad idea in ARAM though, considering that almost every single second of ARAM is a relevant second, and it's 5v5 so you can't use the burst healing to establish much lane dominance.
Here are the corresponding numbers for the Charm:
No selling: 14 minutes 15 seconds
Selling: 4 minutes 15 seconds
Gold Value Per Second: 0.21g/s (12.6g/min)
Here are the corresponding numbers for Spirit Stone, attributing 0 value to the passive:
No selling: 10 minutes 10 seconds
Selling: 3 minutes 5 seconds
Gold Value Per Second: 1.14g/s (68.6g/min)
And lastly the numbers for Philosopher's Stone, taking the gp10 into account:
No selling: 10 minutes 5 seconds
Selling: 5 minutes 10 seconds
Gold Value Per Second: 1.16g/s* (69.4g/min)
*Note that it is assumed that every second is a relevant second.
As always; numbers have been rounded a little to make them a little easier on the eyes.