Being won over may refer to the act of persuading someone or a group of people to change their position or opinion. In the realm of synonyms, the phrase "be persuaded" comes foremost to mind. Other expressions which carry a similar meaning include "be convinced," "be swayed," and "be influenced." You may also use idioms such as "come around" or "see the light." In a more casual tone, one can also say "be sold," "be won over," or even "drink the Kool-Aid." No matter the phrase used, the end goal is to make someone or a group of people open to a new idea or point of view.