General Synonyms:
* Eliminate
* Exterminate
* Destroy
* Annihilate
* Wipe out
* Obliterate
* Root out
* Uproot
* Extinguish
* Abolish
More specific synonyms:
* For diseases: Cure, banish, suppress
* For problems: Solve, resolve, overcome, conquer
* For bad habits: Break, quit, give up
* For harmful substances: Detoxify, purify
The best synonym for "eradicate" will depend on the specific situation and the desired level of formality.