Say Eidolon of Rhetoric is on the battlefield and an opponent has a Cipher card (Hidden Strings for example) encoded on one of their creatures. If that player casts a spell and then deals damage with that creature in the same turn, would they be able to cast Hidden strings? Would it work because the Ciphered card turns additional casts into an ability or would it not work because the cipher ability says "cast"?
They cannot cast the ciphered spell because they've already cast a spell. Rule 101.2 says
The cipher ability allows the player to cast a spell, and Eidolon of Rhetoric says that they can't cast any spells after their first each turn. So, Eidolon prevents them from casting the spell.
From the Cipher rules, as long as the card is encoded on a creature, that creature has an ability that says
So, when the creature deals damage to a player, the player can make a copy of the spell, and then they can cast the spell just like they would cast any other spell except that when they pay the cost, they don't have to pay the mana cost. This spell is affected by Eidolon of Rhetoric, just like any other spell
It doesn't matter whether the instruction to cast comes from an ability or from having priority. They can't cast a second spell since they've already cast a spell.
As the saying goes, "can't trumps can". This means Eidolon of Rhetoric's ability to forbid him to cast more than one spell per turn overrides any instruction to do soCR 101.2.
The triggered ability granted by Cipher is as followsCR 702.98a:
Whenever this creature deals combat damage to a player, you may copy the encoded card and you may cast the copy without paying its mana cost.
You cannot obey the instruction to cast the copy. When faced with an effect with a mandatory instruction that cannot be obeyed, you perform as much as the effect as possibleCR 101.2. However, the instruction to cast is optional in the case of Cipher ("you may"). As such, you are forbidden to chose to take the forbidden actionCR 608.2d.
So all that happens is that a copy of the card is made if you so desire. The copy ceases to exist immediately afterCR 704.5e.
Doing things in the opposite order wouldn't help. If he started his turn by attacking, he'd be able cast the copy of Hidden Strings, but he wouldn't be able to cast anything else in his second main phase for all of the same reasons.
Something that the other answers haven't really emphasized, but might be the key to your confusion: you said
Cipher does not "turn additional casts into an ability." What happens is that when the creature deals damage to the defending player, the cipher ability triggers, and then resolves, and while resolving it gives you the opportunity to copy the exiled card and cast that copy. But you are still casting something. In almost all respects this is just like any other instance of casting a spell: you pick targets, pay costs, and put it on the stack as normal, and anything that affects your ability to cast spells also affects this. There are only two differences: (1) the thing you're casting is a copy, so is not represented by a card, and (2) it's being cast from exile, not from your hand. Neither of those two things make a difference to Eidolon of Rhetoric, though. (Hypothetically, if the Eidolon said "Each player can't cast more than one spell from his or her hand each turn," then it would make a difference, and the spell cast via cipher wouldn't count toward that limit of one.)