General encouragement:
* Inspire
* Motivate
* Stimulate
* Support
* Promote
* Foster
* Bolster
* Empower
* Uplift
* Hearten
* Cheer
More specific synonyms:
* Urge: (To strongly advise or request someone to do something)
* Persuade: (To convince someone to do something)
* Embolden: (To make someone feel more confident)
* Incite: (To stir up or provoke someone to action)
* Prompt: (To cause someone to do something)
Formal synonyms:
* Ameliorate
* Augment
* Exhort
* Impel
The best synonym to use will depend on the specific context and the desired tone.