The phrase "on the edge of tears" describes the feeling of being close to crying but holding back the tears. There are several synonyms that can be used instead of this phrase such as "tearful," "sad," "emotional," "heartbroken," "overwhelmed," "upset," "on the brink of tears," and "drowning in emotion." These words can be used to convey a sense of deep emotional turmoil, whether it be from sadness, stress, or even joy. Choosing the appropriate synonym will depend on the context and the intensity of the emotion being expressed.