The latest possible time you can activate the Eidolon to pay for (part of) the multicoloured spell is during the process of casting that spell, right before paying for it (601.2a-i details the process of casting a spell, in order):
601.2g If the total cost includes a mana payment, the player then has a chance to activate mana abilities (see rule 605, “Mana Abilities”). Mana abilities must be activated before costs are paid.
601.2h The player pays the total cost in any order. Partial payments are not allowed. Unpayable costs can’t be paid.
(This is the reason some mana abilities, in particular random or unpredictable ones, have a restriction to only be activated whenever you could play an instant: It would be very strange to activate them at this stage, when you've revealed the card from your hand, chosen targets and modes, and then hope you can afford it.)
The trigger for a spell being cast comes later (emphasis mine):
601.2i Once the steps described in 601.2a–h are completed, effects that modify the characteristics of the spell as it’s cast are applied, then the spell becomes cast. Any abilities that trigger when a spell is cast or put onto the stack trigger at this time. If the spell’s controller had priority before casting it, they get priority.
At this stage, regardless of how you chose to activate the Eidolon, the mana ability has already been activated, paid for, and the mana received and used, so the Eidolon is in the graveyard, and it will therefore trigger.