General terms:
* Carousel - This is the most common and closest synonym.
* Roundabout - This term is used more in British English.
* Calypso - A less common synonym, often used in the US.
More descriptive terms:
* Spinning horses - Emphasizes the movement and features of the ride.
* Whirl-a-gig - Conveys the energetic, spinning nature of the ride.
* Kiddie ride - This emphasizes that it's a ride for children.
The best synonym for you will depend on the specific context you're using it in.