The technical name in the sports medicine world is medial tibial stress syndrome. At the last talk I gave at Dick Ponds to a great group of runners, I had 2-3 in the crowd with nagging pain to the inside of the shin just above the inner ankle bone. At this point everyone had the pain early on in the run and it would easily go away by the next day. I didn't have the opportunity to check foot shape or Q-angle or shoe wear but I'd put my money on muscle bone irritation as the issue. I heard a fast solution from one of the runners as increasing the angle of the treadmill 1-2 degrees. I have to investigate that but sounds logical. Only problem would be the impact on the knees with the increased angle. My usual go to pathology with shin splints is to check the posterior tibial tendon.
Usually if the arch of the foot shows some collapse, there would be a tugging to the muscle tendon that comes down from the leg and inserts into the "fallen arch". One would think one inch shouldn't matter but after 30 minutes of pavement running, the one inch extra stretch will soon tear somewhere from the insertion of the tendon on the arch or the origin of the whole muscle group to the inside of the shin. Throwing an arch support into the shoe is easy, just have to make sure there's enough room in the shoe and that other parts of the foot won't be too compressed in an already "glove fitting" shoe. My friend Bob Weil, DPM, is great at constructing orthotics for athletes. http://sportsdoctorradio.com/index.html
In fact, in many cases, Bob has created orthotics and changed not only the angle of the foot arch but also the knee thigh and hip angles (also know as the Q-angle) to significantly improve friction issues at the knee or hip. We probably wouldn't hear much of these injuries a few decades ago in the 30-40 year old age group but now with the advent of the running clubs, 5k's and multiple endurance sports....overuse injuries are more rampant in the Boomers and X-gens.
The suggestion for cheap cure is throw an arch in (if it works you may want to invest in the permanents made by a podiatrist). Kinesio taping is coming back but some therapist/trainers aren't great at taping, it's a form of long lasting tape application that uses tugging on the skin to remind an athlete to turn out the foot and create an arch. Of course training the muscle group to be more adept at the 30 minutes of stress or increasing the pliability with hydration, omega 3 fish oil or deep tissue massage may help. Maybe just buying one of the running shoes built for "pronators" will do the trick. Finally, I always warn women about the possibility of stress fractures that masquerade as on and off pain. Backing off on the exercise will give the fastest results but this is usually a bargaining chip I save until later for my new athletes.