General synonyms:
* Draw out
* Evoke
* Bring out
* Extract
* Obtain
* Trigger
* Produce
* Generate
* Induce
* Prompt
* Solicit
* Uncover
* Reveal
More specific synonyms:
* Invoke: This implies calling upon something or someone for help or inspiration.
* Inspire: This focuses on generating a feeling or idea in someone.
* Unleash: This implies releasing something that was previously held back.
* Squeeze out: This suggests a more forceful or demanding way of getting information.
* Ferret out: This implies uncovering something hidden or secret.
The best synonym for "elicit" will depend on the specific situation and the intended meaning.