Running shoe weight is something that should definitely be factored into your decision making, along with the other shoe features, when looking for the optimal shoe for you. As a general rule regarding weight, lighter is better, but this definitely does not mean you should rush out and get the lightest shoes you can find! If you could have the exact same features in two otherwise equal shoes in every way (comparing ”apples to apples”), the lighter weight shoe would be the shoe of choice, however, certain features such as stability, support, cushioning, durability, and proper fit may necessitate a heavier shoe for one person than another depending on variables such as the individual's body weight, biomechanics, foot-type, running surfaces, mileage, running pace, physiology (such as bone-density), and injury history.
The question then becomes, how light a shoe can you find that has all the features you need? Another question might be, are there any features worth giving up that would be offset by the advantages of a lighter shoe? One personal example I can give is a nagging hamstring injury I suffered with off and on for years. Some factors contributed to this condition that I could work on, including my run technique and hamstring strength, however, my hamstring symptoms have resolved since switching to lightweight shoes.
From a biomechanical standpoint, having a lighter weight (a lighter shoe) at the end of a swinging pendulum (your leg) allows for less work out of the hamstring muscles and tendons to swing the leg. Also, one can run with a higher cadence with less weight to swing through the air, which leads to improved running technique, which leads to better efficiency and less injury potential. In my situation, I sacrificed some cushioning in the past to obtain the reduced shoe weight, and as a result, suffered through two stress fractures (having low bone-density didn't help). I then found a pair of lightweight running shoes that had more cushioning and the support I needed, which along with upping my vitamin D intake, will hopefully prevent future stress fractures.
It must be stressed that lighter weight running shoes generally tend to be less durable, have less torsional stiffness, fewer stability features, and less cushioning than their heavier counterparts, however, running shoe companies are constantly introducing shoes with
lighter materials that also provide superior support, cushioning, and durability.