what should a diabetic eat when training for a half marathon?

First of all, I would like to make a disclaimer. There is no known cure for diabetes and I am not suggesting that there is one.
However, with that said, there is a ton of research and documentation on various nutrients, supplements and even diet changes that can benefit and help if you have diabetes.
Although there may not be a "cure" there are things that you can do to improve the condition, depending on the type of diabetes that you have and how severe it is.
Obviously there are certain food to avoid altogether if you really want to do your body good. These include refined sugar products such as candy, ice cream, cakes, cookies, pies, pastries, soda, jam and the like.
Even natural sweeteners like honey, maple syrup and molasses should be used sparingly.
Drink plenty of water if you can and try and consume 8 glasses of pure water daily. An alkaline diet is best including lots of fresh vegetables, especially raw vegetables and salads. You can eat as much of theses healthy foods as you like as cooked foods tend to cause more of a rise of glucose in the body than do raw fruits and vegetables.
Eat plenty of fruits such as grapefruit, bananas, granny smith apples, figs, cranberry, blackberry, pomegranate, kiwi and citrus fruits.
Eating a high fiber diet is ideal since it lowers the body's need for insulin. Make sure you are including foods such as barley, oatmeal, almond meal, dried beans, kidney beans, cooked black beans, peas, cereals and chickpeas in your daily diet.
It is extremely important to monitor your glucose level if you are a diabetic or have pre-diabetes symptoms. Here is a guide that will help you measure the proper levels:

Upon waking up (and before breakfast), your levels should be between 80 and 120. This is considered to be a healthy range.

Before meals your levels should also be between 80 and 120.

Two hours after you eat meals, your levels should be 170 or less.

Before lunch, it should drop back down to 80-120.

Before you go to bed, it is ideal to be between 100 and 140.

At 3am (while sleeping), it is ideal to be between 70 and 110.

While fasting, it is ideal that you stay in the 70 to 100 range.