I think its normal. It's an old car, the wires are probably old and developed more resistance, so your alternator is just not producing enough juice. In any case, it has nothing to do with the battery at all, once the car is started it doesnt use the battery power at all, in fact it does exactly the opposite - uses power generated by the alternator and charges the battery.

In fact this is fairly normal for most cars, it's just that most people do not notice it. And your moms RX300 is 10x brighter because it most likely has HID system with high power ballasts. You can buy an HID kit for your car for maybe $350.

Another thing you can do is buy a more powerful alternator, but
1) It will most likely eliminate the problem with dimming headlights
2) It will not make your lights brighter however
3) You might lose 1 or two horse powers

Hope that helps.