Purely from marketing point Lucangosteen sounds too long and complex. Limoncillo sounds somewhat "lemony" and Spanish sounding. Asians may have problem with pronunciation. Mexican Garcinia no go because ordinary people don't know what garcinia means. Lucosteen might be the best, but still gives no suggestion to people on fruit qualities. Mexican mangosteen or even American mangosteen (for marketing in countries like Taiwan) might be the best even if it's two words and long name. Word mangosteen immediately associated with high quality and for many it would give good hint.
I agree whole heartedly on all points, and in addition as others have said limoncillo would cause too much confusion with other fruits that also have that name including my beloved spanish lime. Which is also a terrible name, but at least from a marketing standpoint it gives a few hints about the fruit. It has a citrus like tang, a lime like appearance too. It's popular in spanish speaking countries, I'm guessing it might have been also spread by spanish settlers back in the day. Most of all the words "spanish" and "lime" are familiar to westerners or english speakers.
That's why I like American Mangosteen, it's simple and easy for people to grab. And like durianlover said, the name mangosteen already has merit. I love Mexico and I don't have negative connotations towards the country or the things they produce, but I could see how the rest of the world might have some stigma attached to that name. The last thing I remember in the news about Mexico was the cartel raising the prices of limes. Sadly I think leaving that name out would help this fruit be more marketable. I think it would be great though if this fruit could become real popular and take off. Maybe one day we could find this fruit in farmers markets here in the states. It has good potential since it appears we can grow it here too.
Another name I though of was Lucillo. Not descriptive, but gives an ode to Luc and sounds nice. Grand garcinia? Magnum Mangosteen?