General:
* Joy
* Pleasure
* Contentment
* Delight
* Gladness
* Cheerfulness
* Elation
* Bliss
* Euphoria
* Rapture
* Exuberance
* Serenity
* Peace
* Well-being
More specific:
* Joyful: Expressing great happiness.
* Exultant: Feeling or expressing great happiness or triumph.
* Jubilant: Feeling or expressing great happiness and triumph.
* Merry: Cheerful and lighthearted.
* Giddy: Feeling lightheaded and excited, often with a sense of unreality.
* Enthusiastic: Having or showing intense and eager enjoyment, interest, or approval.
* Elated: Feeling very happy and excited.
* Thrilled: Feeling very excited and pleased.
* Ecstatic: Feeling or expressing overwhelming happiness or joy.
* Fulfilled: Experiencing a sense of satisfaction and achievement.
* Content: Satisfied with what one has.
* At ease: Relaxed and comfortable.
More abstract:
* Fulfillment: The state of being happy and satisfied with one's life.
* Well-being: A state of good physical and mental health.
* Prosperity: Success and good fortune.
* Flourishing: Thriving and growing.
The best synonym for happiness depends on the specific context and the shade of meaning you want to convey.