I've poked around inside my i931x but only because a pet hair made its way into the LCD screen which drove me nuts. I noticed the amplifier part of the boombox uses a single IC chip to drive the speakers. Of course this is the least expensive and lowest quality way to amplify. The speakers are huge looking on the front but sport a small magnet on the back. I noticed a small filtering capacitor seperates the frequency for the tweeter. 

If you were to upgrade the speakers and beef up the two way speaker crossover circuitry using a higher grade passive system, that would be a good start.

I had a vision about how to increase the output power but it's kind of cheesy and I'm not sure it would even work. I was thinking of using the existing amp output to the speakers since its a harness connection to the board. Run that into a speaker level to line level converter or just solder into the low line output of the headphone jack.  Feed that into a small car audio aplifier circuit board (maybe 25x2 in Watts) mounted to the back cover of the stereo and then run that into aftermarket speakers.

Several issues with this would be space and power issues. They make car stereo circuits which are quite small and they're designed to run on 12 volts. Well voltage we're good on as we have 15 volts in power but amperage is another. We're talking about mA when it comes to batteries. If you can find a higher wattage amp that will run on the power requirements this would be a good mod.

I don't think I would want my box to be the guinea pig      :-)

The circuitry that's in there is very small boards. I remembered thinking that the body of the unit was way larger than the internal parts.