I guess I would need to understand your idea of worthwhile brightness. I presume they are more or less decorative? Are you looking at this one?:
http://www.amazon.com/dp/B00JF0589E?psc=1
Cloning that shouldn't be too difficult and controlling it with the computer should be easy enough, though not as in-expensive as you are thinking. Not that easy to get I/O right out of a computer that is software controlled.
Here is this little guy. Basically what you could do for a simple solution is to replace your manual input on the switches on the remote with a digital version. Basically wire up the I/O on the board to a logic circuit that will provide a normally open or normally closed output to each switch (depending on how that remote is wired up) This would leave the remote in the loop though so it would still be IR to the control box, but it would be out of sight and you would be able to control it from the computer.
https://www.sparkfun.com/products/762?gclid=CKbspMfx58UCFQcpaQodnacAtw
Basically this would have a logic diagram implemented in code to trigger the I/O to act as a giant selector, one unique combo for each button. If that makes sense. The commands would be via a virtual serial port. You can then write a simple C/C++/C#, or whatever really, program in something like Visual Studio Express or Eclipse to send the serial commands. (Can't find a decent wiring example to use for this, you can look at 7-segment displays to get an idea of what I am talking about on the physical side)
The other option would be to see what the other end is doing and take the remote out of the loop completely. Not sure how they are controlling the LEDs in there, so it might require the complete replacement of the LED driver circuitry, which wouldn't be cheap in terms of hardware. Then you would have to create a hardware interface to the computer, possibly using the board above, and again write the software.
I think you might have an easier time of it with a basic arduino board and an RGB Shield. You can then just buy the raw RGB LED ribbon without all the inbetween stuff.
https://www.sparkfun.com/products/12757
http://www.robotmesh.com/rgb-led-strip-driver-shield-v1-0?gclid=CM_At6r158UCFVM1aQodgwYAlQ
You should be able to write an interface program between the computer and the Arduino that will let you do it. Also expandable and you can use it for all kinds of things including some of the bluetooth and WiFi options I mentioned before.
As for the cheapest possible method, use it as is and deal with the remote.