The phrase "stay on" refers to remaining in a particular place or continuing with a particular task. There are several synonyms for this expression. For instance, one could use the words persist, endure, remain, persevere, or linger to mean the same thing. Additionally, the phrase "stick with" can also replace "stay on" when referring to continuing with a task. Moreover, the phrase "keep going" can also be a substitute for "stay on" when referring to an individual or group's persistence with a task. In conclusion, there are several phrases or words that can replace "stay on" to express continuing with a particular task or remaining in a particular place.