You need to finish the side quests given by the Blue Pharmacy, similarly to how magic potions to refill your mana are unlocked after you finish their quest.
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Checked the script, and the requirements to unlock the elixir are:
-Finish all the blue pharmacy quests (Last one is the mushroom one that unlocks magic potions to restore mana)
-Reach the point of the story where Bell learns about the familia's debt, shortly after acquiring the new familia home.