It is very difficult, but not impossible. Generally, Boards for Correction of Military Records upgrade punitive discharges as part of clemency, but they do this under very unique circumstances. They will probably look at your age at the time, the seriousness of the offense, and whether your misconduct outweights all other positive military service, among other factors. Many attorneys here offer free consultations. I recently dealt with a comparable issue. If you would like to schedule a free consultation to discuss this, feel free to contact me.
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It would be very hard. I assume your appeal is done - probably for well over a decade. The time frame for petitioning the Naval Discharge Review Board is usually 15 years and for the Board for Correction of Naval Records it is usually 3 years. You can ask for leave to file beyond those time frames, but they can say no on that basis. If you want to take the next step and file - you can call me for further discussion.
If you said "Army" I'd say, you might have a shot. The army likes to consider where you've been and where you are now. The Marines have repeatedly made it clear that they simply do not upgrade. While I have been successful with the Army, I have never had anything upgraded by the marines. I don't know if anyone on this site has had a different experience with the Marine Board, but mine have not been positive.
I am not going to give you a "legal" answer because the ones provided are excellent. Nearly two decades have passed. The best thing that you can do to "clear your name" is continue living with the integrity expected of a Marine. Show that your conduct was an aberration and does not define who you are now.
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It is an uphill battle- but not impossible. In order to have your discharge upgraded you have to apply to the Naval Discharge Review Board (NDRB) in Washington, D.C. I currently have a package before the board. Additional information is available on my website. Feel free to call me for a free consultation.