See this post about the likely Egyptian origin of Moshe's name. R. Zvi Sobolofsky, in this week's TorahWeb essay, writes the following about his retention of that name (link):
His name “Moshe” tells us a lot about his personality. One would have expected Moshe not to use his Egyptian name given to him by the daughter of Pharoh. Yet Moshe insisted on keeping this name as an expression of gratitude to the woman who saved his life as an infant and raised him. Moshe’s commitment to showing gratitude to those who assisted him in times of need is evident from his approaching Yisro before returning to Mitzrayim. Hashem had commanded him to return immediately to begin the process that would free entire Jewish people from slavery. Yet before going, Moshe approaches Yisro and tells him of his plans to leave. Chazal comment that Moshe was not only informing him that he was going, but he was also asking him permission to leave. As critical as his mission was, it was inconceivable to depart from Yisro without his permission. Yisro took him in when he was in need and provided for him for many years. As important as saving the Jewish people was, Moshe would not act in an ungrateful manner to one who had helped him. It is this commitment to hakoras hatov – expressing appreciation to others – that Moshe personified.