On the basis that "Full HD 1080p" is the term used (appears on the splash screen) when my generic black box media player (Avedia) boots up, then I'm guessing yours may possibly be one of these clones which are based on the
Realtek1073 chipset. Usually, but not always, the on-screen UI for these boxes will look a lot like this:
http://news.kikil.com/images/2009/11/f6 ... en.jpg.jpg
There are dozens of these on the market under different names - unfortunately I do know that different brands can use different remote control setups depending on how the manufacturer configures the Realtek firmware for the box.
Here's a link to the RM file for my player: (NEC2 device 0)
https://www.hifi-remote.com/forums/dload ... le_id=8206
and here's one for the
Patriot Box Office which is one of the more popular brands using this chipset: (NEC1 device 73)
https://www.hifi-remote.com/forums/dload ... le_id=8255
and the thread where that is discussed:
https://www.hifi-remote.com/forums/viewtopic.php?t=12022
Neither may work for yours - but one just might, and since you've had no response I thought I'd throw this in as a possible starting point for you, if you're still looking.
If that doesn't help - then either buying a cheap JP1 learning remote and cable or finding someone who has them would enable you to learn and decode the signals from your existing remote control very easily, both now and for any future devices you acquire - this would, by far, be my recommended route for you.
There's also this:
http://betaremotes.slingbox.com/ which might help, but I don't have a slingbox so I can't tell you if it works or not, or how to use it...
These Realtek 1073 players (
if yours is one of these things... 
) are basically linux boxes, (the "firmware" is pretty much the whole mini linux OS) but the main "guts" of their operation is contained in a single monolithic binary executable which the various manufacturers can customise using an SDK supplied by Realtek. This compiled executable contains the display drivers, media player and, you guessed it, remote control functions.
There's been some limited success hacking these to run each others' firmware - and when they do that the remote control codes alter too - the PBO forums have lots of threads on this, e.g:
http://www.patriotmem.com/forums/showpo ... ostcount=5
http://www.patriotmem.com/forums/forumdisplay.php?f=24
but that's
not for the fainthearted, since you can "brick" your player doing unofficial things like that...
