So first off, the scenario.

"My players ended their most recent session with them entering a creepy room and, after doing something, accidentally summoning about 7 gibbering mouthers. The room isn't actually all that big, so many of the players are inside of the AOE effects of most of the mouthers. Their effects require saves when a creature starts their turns inside of an AOE."

My Question: If a player starts their turn within the AOE of multiple creatures and said AOE forces the player to make a save at the start of their turn, do they have to make the save multiple times(one for each AOE they're inside of)?

Generally, yes. Though for the mouthers, failing one save is sufficient to induce the effect. These sorts of things can seriously slow down combat, so for the sake of the encounter, I might just rule that there's one save to make, and adjust the DC up by the number of living mouthers.

Your tooltip didn't work. If you could fix it or just have the name of the creature so we can see the statblock, it would be easier to see what's happening.

Fixed it thank you for letting me know.

First, two or more gibbering mouther should attack each other.   They are "forced to consume everything in reach."  Being insane, I cannot think of much that would prevent this, not without redesigning the creature significantly.

Second, they are NOT immune to the gibbering of other mouthers.   So even if they did not intend to attack their allies, they themselves would have to make the saves against the other gibbering mouthers if they were in range.  Again, I cannot think of much that will stop this, without a LOT of house ruling.

It makes more sense to have one giant gibbering mouther, with higher DCs.    That said, if you were to imprison multiple gibbering mouthers, locking each up 21 ft away from each other, that would leave an area that would overlap, possibly as three at any one given time.  

In any case, I would rule that difficult terrain can not be made worse, so the strength save is once per turn.  The gibbering should have multiple saves but as they are at least 20 ft away from each other, I doubt anyone will have more than three saves in any one location.

Technically, I think it should be separate saves. However, personally, I'd just have them do one roll per type of save. If they succeed, great. If they fail, have them roll a percentage. Reduce the total that would have been done by all the monsters by that much. Having each player do 14 saves would slow things down too much.

Quote from Okashido >>

So first off, the scenario.

"My players ended their most recent session with them entering a creepy room and, after doing something, accidentally summoning about 7 gibbering mouthers. The room isn't actually all that big, so many of the players are inside of the AOE effects of most of the mouthers. Their effects require saves when a creature starts their turns inside of an AOE."

My Question: If a player starts their turn within the AOE of multiple creatures and said AOE forces the player to make a save at the start of their turn, do they have to make the save multiple times(one for each AOE they're inside of)?

 Heart is correct for both effects (I'm only chiming in in case their post was interpreted as only referring to one of the 2).  Anyone starting their turn close enough to multiple mouthers has to make a strength save and a wisdom save per mouther, and any one failure is enough for that specific effect to take hold, so any one Str save failure inflicts the Str failure effect, and likewise for Wis.

Quote from Mog_Dracov >>

In any case, I would rule that difficult terrain can not be made worse, so the strength save is once per turn.

I like this analogy. If the character resists one Mouther, I do not see why the character would be susceptible to the others if they all do the same AoE thing to cause the effect. If they each directed an attack at the same person, that would be a different matter where each are doing it on their individual turns.