Another name with next to no info readily available, there are a few possibilities for meaning here. Rhondine could be a rare French variant of Rhonda or a created English variant of Rhonda, meaning "rose" in Greek, "good lance" in Welsh. However, drop the letter h and we have Rondine, meaning "swallow." Rondina can also be found as a first name and mainly a surname. There is an Italian opera titled La Rondine, "The Swallow" in English. Interestingly, rondinella is the name of an Italian wine grape variety.
At least 11 people in the U.S. have Rhondine as a first name, possibly about 20 total, and maybe only 1 has Rhondina as a first name. There are some early records, one in 1888. My guess is that this name came about in several different ways - possibly a vintage trend where familiar names were elaborated upon, a misspelling of an original name, given in reference to the opera, or the transfer of a surname to first name. There are about 50 people named Rondine in the U.S. and about 20 Rondina. I've also seen a small handful spelled Rhondeen and Rhondean, which are probably smush names such as Rhonda + Eileen or Rhonda + Dean.