Doing some strength training is an important part of a training program. Strength training helps to correct muscle imbalances created by running, which in turn, helps to prevent injury. It also builds muscle that can allow you to handle a greater workload. However, it can also take valuable energy and time away from running training, and actual running is, without a doubt, the best way to get better at running. Additionally, putting on a lot of muscle weight in areas that don't directly aid running equates to dead weight that you have to drag through your races (and while a certain amount of body fat is crucial for fat-soluble vitamin absorption and maintaining body function, body builder arms are not). Therefore, the key to effective strength training is, like any other aspect of running, specificity of training.
My final project for my physiology class is writing an exercise prescription for myself. That required figuring out what my strengths and weaknesses were, and mapping out a plan to improve the weaknesses. For the strength training portion of the project, unsurprisingly, my leg strength was excellent, but my biceps and triceps strength was decidedly average. However, would it really be a good use of my time and energy to stand around curling 20-lb weights three times a week? We all need functional strength. There's no debating that. If you can't carry your groceries from your car to your house, you need more arm strength in order to function in society. Similarly, if your arms can't swing through 26.2 miles, you're not going to be a very good marathoner. However, there's a huge difference between being able to perform functional activities and having Arnold Schwarzenegger arms. This is part of why I cringe when I hear that someone is going to do P90X during the training cycle for their next race. P90X is great for all-around strength. However, the program in its entirety is terrible for true running training (but if you know how to pick and choose exercises such that the time and energy drain from the full-program isn't there and you're also only doing the exercises that best supplement your running, it can be useful).
So now the problem becomes determining what kind of strength training you should be doing. In Lore of Running, Noakes recommends quadriceps strength for increasing endurance and injury prevention. And if you're doing quads, you also need to work hamstrings, since they are the opposing muscle group. And in Daniels' Running Formula, Daniels suggests core strength to improve balance, body control, and running economy (core does not only include abs, but the entire core: rectus abdominus, internal and external oblique, transversus abdominus, and a whole host of back muscles that I won't get into here).
But let's face it: that's really basic, but it's also what you have to put in such a general case book. Similar to the basic training programs in those books, those programs can be made more effective by personalizing them, such that they isolate your personal weaknesses and work to minimize them. Here's an example for you: there are several parts of the body that, when strengthened, will benefit rock climbers: back, shoulders, arms, forearms, hands, etc. However, to try to thoroughly hit every single muscle group would take forever, and little time and energy would be left to actually climb (and as any reasonably strong person who has tried rock climbing knows, strength alone doesn't get you up the rock). So, my friend Luke, an avid climber who has had shoulder problems in the past, directs the bulk of his strength training focus on his rotator cuff (specifically the muscles that do shoulder flexion and extension). That's not to say that he never does strength training for other important muscles, but to spend the same amount of time on his finger flexors as his shoulder would be silly, since it would take away from his climbing time. Similarly, my friend Molly, who is an elite distance runner and therefore really needs to focus her training appropriately, spends a lot more time on her hip stabilizers than her quads. That's not to say that this kind of training is fun. I hate working the little intrinsic and extrinsic muscles of my feet. Frankly, it's boring (and I don't get that pump where I can walk in front of the mirror and flex my little T-Rex arms, but that's another story). However, it's probably the most important strength training that I can do.
The last point is that you want to be careful not to overdo it. Since these are your weaknesses, you're also working with weaker tendons. Structural changes take a long time (longer than cardiovascular changes, for example, which is why it's so easy to do too much too soon when running, and injure tissue that's not quite ready for the increased volume or intensity). So go slowly and allow your tissues time to adapt. No one gets stronger from doing hard workouts. You get stronger by recovering from your workouts.
Just like there's no need for sprinters to do 20 mile overdistance runs, there's no need for marathoners to slog through hours and hours of strength training every week. The key to effective training is isolating your weaknesses and imbalances, figuring out how they relate to running, and focusing on those exercises that will most benefit your training.