At first glance, though I haven't heard of this being done, it sounds like a reasonable approach, similar to the way that a solar hot water system or a drain water heat recovery system preheats the water before sending to the hot water heater. By storing the incoming cold water indoors and letting it warm up to room temperature, the "Delta T" -- the difference between the incoming cold water temperature and the temperature that the hot water heater has to heat to -- is reduced. So the heater has less work to do.
But you asked whether it would keep your water hotter LONGER, by which I'm guessing you mean whether you'll run out of hot water less often. That's a different question from whether it would reduce how much the water has to be heated. Assuming you're talking about using a conventional hot water heater (versus a tankless heater), the limitation on how much hot water you have available really has to do with the capacity of the tank. Once you have used what is heated in the tank (do you have teenage kids?), even if the water is pre-warmed, it still has to refill and heat the water. That cold jolt when the hot water runs out about may be a little less nasty, but it will still happen.
With tankless or on-demand systems, the equation is different. Whether you get that unwanted morning startle has more do with the flow rate of the heater. In this case, pre-warming the water may indeed help some.
If you want to get more analytical about this, what you'd appear to be gaining is energy savings by cutting down that Delta T. But even that may be illusory since that heat has to come from somewhere. In other words, in the winter, you have to heat the house in order to have a warm home for that preheat tank. It's another example of the thermodynamic version of "there's no such thing as a free lunch." It's not as true in the summer, by the way, since you're probably trying to cool the house anyway. In those warm months, you may indeed be able to eat your cake and have it, too.
By the way, a better first step may be simply to insulate the hot water tank. There are kits for this. Unless you have a good new heater, they tend to be pretty inefficient, losing a lot of their heat while just standing around waiting to be used.