Why was this system put in place originally?
There seems to be no canonical answer. From the Wikia "The Chain" article ("Vampire Elders" had no info):
... For example, Underworld: Evolution shows that all three Elders were awake at the same time in 1202 AD, witnessing William Corvinus being captured by the Death Dealers, the original Werewolf having just ravaged an entire village and turned several dead civilians. Marcus, who had been promised by Viktor that his brother would not be harmed, was betrayed by Viktor and Amelia to keep William and in turn Marcus himself under control.
However, at what point the Elders agreed to hibernate when not reigning, whether at the Chain's inception or an added measure included centuries later, is also not known. If the latter, the hibernations may have been devised by Viktor and Amelia with the intention of keeping Marcus and William separate from each other forever.
Another alternate theory (mine) could be that it was done to avoid leadership struggles. Victor believed that he couldn't kill Marcus due to bloodline issues, so couldn't just eliminate him, and Marcus was weaker so couldn't eliminate Victor. Since they all read Star Wars extended Universe, they realized from the pre-Rule-of-Two-Sith example that the leadership struggles are bad, and therefore decided to only have one elder active at a time.
Viktor tells Selene at one point that "these rules are in place for a good reason and they are the only reason we have survived this long". The rules are part of a "covenant", and form a "chain" of elders' rule.
I first thought that vampires, while "immortal", may have some sort of maximum lifespan, and that being in hibernation further extends their lives. However, Underworld: Evolution shows us the very first immortal, Alexander Corvinus, in perfect condition after over 1500 years. In that movie, it's learned that age only makes an immortal stronger. So that's out.
The second theory, that the elders hibernate so there is only one elder making decisions for the vampire species, and so less chance for conflict, also holds some water. However, in the first movie Kraven basically has control of the European coven while Amelia is off managing her own affairs in America (an irony, as Amelia's actor Zita Gorog isn't fluent in English). I'm sure Kraven is at some level subject to Amelia's decisions, but nothing ever comes of it.
The third theory, and the one I agree with most, is that with two of the three vampire elders in relatively safe quarters at any given time, more of the Death Dealers can be used to hunt the Lycans rather than to protect all three leaders in various locations around the world. That would mean the Chain was formed after Lucian's escape, as a result of it. It makes a certain sense, especially considering the myth that if the elders die, any vampire they created will die. Facing that possibility, you would want as much protection as you could give one of them, and then the other two would be pretty safe in the well-guarded castle/mansion. Despite the Lycans not seeming to pose a very serious threat to the existence of the vampires as a whole for the nearly 600 years since Lucian's forces were apparently defeated, this theory makes the most sense overall.
I was under the impression that the elders were significantly older than the other vampires. Likely as a result of the war with the Lycans. As older vampires they need to hibernate to avoid going insane (or it is simply a requirement of being an elder). When the elders realized they would need to hibernate, they setup a system so that the leadership would never be vacant.
Differing opinions may also be the reason. In order to avoid strife between vampire factions they decided that only one elder should be awake at a time. This eliminates any division that may occur. As a result they would be a stronger unified force against Lycans (or any other enemy that crops up).
There was a similar hibernation requirement in the Anne Rice Vampire Chronicles. Vampires eventually get to a point where they need to hibernate for x number of years. A deep uncontrollable sleep overcomes them.