Root crops such as Potatoes and Beets benefit from mint mulch, but there is no reliable indication that they derive any benefit from being grown with it. Carrots benefit from the mints repulsion of carrot root fly. Onions and mint intercropped has proven beneficial to both. 
Agrarian Folklore states that mint is a good companion plant for
beets, broccoli, Brussels sprouts, cabbages, cauliflower, eggplant, kale, kohlrabi, lettuce, peppers, squash and tomatoes.In some instances it improves their vitality and flavor, in others it protects against pests.
Brassica / Cruciferous vegetables such as cabbages, broccoli, cauliflower, Brussels sprouts and kale [as per lore] are said to perform better when planted with mint.
It does not go well with most other aromatic herbs, and do not mix different varieties of mint together as they will cross breed generally producing undesirable results.
1. "It was shown more recently, however, that even highly aromatic companion plants, such as ... mint (Mentha piperita L.), used frequently by organic growers to protect their valued crop, were not 'repellent' ... Instead, they produced their effects by disrupting the normal 'chain' of behaviours" - The influence of host and non-host companion plants on the behaviour of pest insects in field crops
2. Companion Plants and How to Use Them by Helen Philbrick , Richard B. Gregg, Herbert H. Koepf.
3. High economic returns from companion and relay cropping of bread wheat and menthol mint
Inter-cropping of onion in menthol mint for higher profit under subtropical conditions of north Indian plains.