Text:
When Idol of Endurance enters the battlefield, exile all creature cards with converted mana cost 3 or less from your graveyard until Idol of Endurance leaves the battlefield.
{1}{W}, {T}: Until end of turn, you may cast a creature spell from among the cards exiled with Idol of Endurance without paying its mana cost.
Bestow {5}{U}{U} (If you cast this card for its bestow cost, it’s an Aura spell with enchant creature. It becomes a creature again if it’s not attached to a creature.)
Flying
You control enchanted creature.
Enchanted creature gets +1/+1 and has flying.
If I have Hypnotic Siren in my graveyard when I play Idol of Endurance and Hypnotic Siren is exiled as part of Idol of Endurance's enter the battlefield effect then I can pay 1W to play Hypnotic Siren with out paying it's mana cost. Does that include the bestow cost? In other words can I pay 1W and play Hypnotic Siren as an Aura card and take control of another creature?
No, you cannot. Bestow is an alternative casting cost, and so is the "without paying its mana cost" from the Idol's effect. You can only apply the latter to cast the Siren from exile here, so it can only be cast as the default creature version.
118.9. Some spells have alternative costs. An alternative cost is a cost listed in a spell's text, or applied to it from another effect, that its controller may pay rather than paying the spell's mana cost. Alternative costs are usually phrased, "You may [action] rather than pay [this object's] mana cost," or "You may cast [this object] without paying its mana cost." Note that some alternative costs are listed in keywords; see rule 702.
118.9a. Only one alternative cost can be applied to any one spell as it's being cast. The controller of the spell announces their intentions to pay that cost as described in rule 601.2b.
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Text:
When Idol of Endurance enters the battlefield, exile all creature cards with converted mana cost 3 or less from your graveyard until Idol of Endurance leaves the battlefield.
{1}{W}, {T}: Until end of turn, you may cast a creature spell from among the cards exiled with Idol of Endurance without paying its mana cost.
Hypnotic Siren
Text:
Bestow {5}{U}{U} (If you cast this card for its bestow cost, it’s an Aura spell with enchant creature. It becomes a creature again if it’s not attached to a creature.)
Flying
You control enchanted creature.
Enchanted creature gets +1/+1 and has flying.
If I have Hypnotic Siren in my graveyard when I play Idol of Endurance and Hypnotic Siren is exiled as part of Idol of Endurance's enter the battlefield effect then I can pay 1W to play Hypnotic Siren with out paying it's mana cost. Does that include the bestow cost? In other words can I pay 1W and play Hypnotic Siren as an Aura card and take control of another creature?
Since Idol specifies that you may cast the spell by paying an alternate cost, you don't get the option to use another alternate cost.
Idol of Endurance's ability cannot be used with bestow, mutate or the Adventure (instant/sorcery) half of Adventurer cards.
Bestow:
Adventure: Casting the Adventure half would result in a non-creature spell being cast in violation of Idol's ability rule text.