Eggs, milk products, fish, and meat (in this order) have the highest biological values, so it goes without saying that your dog would do well to eat these. What about the other foods, though?
Fruits and veggies that offer high amounts of phytonutrients (also called phytochemicals) can benefit in many ways from working to reduce inflammation to providing antioxidants that help fight cancer. All fruits and veggies provide some of these good things, but some are more powerful than others.
For example, a study in the Journal of the American Veterinary Association (Evaluation of the effect of dietary vegetable consumption on reducing risk of transitional cell carcinoma of the urinary bladder in Scottish Terriers :227 (1):94-100) concluded that a preventive effect was seen in dogs eating green, yellow, or orange vegetables three or more times weekly.
Best green veggies: broccoli and kale for their anti-inflammtory properties; kale (again) and spinach as a liver cleanse. Foods in the brassica family (cruciferous veg like broccoli, cauliflower, cabbage et al) can affect healthy thyroid function if overfed, and many of my clients are concerned about that. We don`t feed enough of them to dogs to make much of an impact on the thyroid, and ideally you`ll be rotating these great greens with some of the foods below, so I wouldn`t lose sleep over it.
Most under rated veg: Carrots! 
Available as orange, red, yellow and purple, all are powerhouses in their own right. They are beneficial to healthy heart function, and your mother was right (again!) about eating carrots to support eye health. As to cancer, studies in people (ok, not in dogs, but certainly better than looking at in-vitro, or rats) show that colon cancer risk is greatly reduced when carrots are consumed. Nobody knows if this same benefit applies to other forms of cancer…but I`ll hedge my bets.
Best fruits: Blueberries, raspberries and blackberries for their anti-cancer and anti-inflammatory benefits from anthocyandins (and a very long list of other flavanoids and phytonutrients). These delicious morsels provide phenolic compounds that prevent blood vessel cancer growth, work as antibacterial, antioxidants, and promote capillary strength.
Caution: introduce fruits (especially berries) and veggies in small amounts at first. Consuming too much leads to loose stool whereas increasing the amount slowly over a period of time works well for most dogs.