I drive a 1999 Chrysler 300M, V6 3.5L engine. I am about to start a water pump replacement repair. I've discovered that the water pump is driven off of the timing belt, rather than either of the two fan belts on the front. This obviously complicates the repair quite a bit. What is the advantage to running it off of the timing belt? Why is it designed this way?
Engine safety. If you lose an accessory belt driven water pump, you're likely to keep driving, thinking the "oh, I just don't have an alternator" while you're busy cooking your engine beyond repair (normally with no temperature notification/change, if anything it'll read cold). When the water pump is on with the timing belt (or geared to crankshaft as is occasionally the case), if the water pump seizes and tosses the belt, the engine will shutdown. Inconvenient, but rarely dangerous, and saves a very expensive repair.
I think the main reason for this is convenience. It's an easy place to run the water pump. If you ran it out to a fan belt, it would be in the way of the timing belt while doing it, or it would be a really awkward mess trying to work around it. The second reason is for compactness. With the water pump stuck out of the way, it physically makes the engine shorter, which means it will fit in the hole which is the engine bay a little easier.
Most overhead cam engines which run a timing belt use this method to propel the water pump. It has been done this way for a long while.
On a separate note, since you are in there replacing your water pump, replace the timing belt and any tensioners which may be there with it. No sense in having to take it apart in another couple thousand miles to get it done. This would be the same reasoning behind getting the water pump done when you do your timing belt. It's just cheap insurance and beats having to do the same work twice.
Historically the water pump was at the front of the engine in close proximity to the vehicle radiator, when vehicles were mostly rear wheel drive. There is no real advantage with todays vehicles but they continue to be at the front of the engine. Using the timing belt to drive the water pump lends itself to compactness at the front end of the engine. Some special purpose vehicles have remote or supplementary water pumps.
Having snapped alternator and water pump belts in the past, I feel the most likely reason is engine safety. If it's a non-interference engine, having the timing belt drive the water pump ensures that the engine safely shuts down in the event that the water pump belt breaks. That way you're just replacing a timing belt instead of rebuilding the engine after it overheats.