Nike Frees have a very soft, unstructured upper and relatively thin sole. Also quite light.
Most "barefoot" or minimal shoes are great for gym due to being completely flat and great for travel due to very thin soles and unstructured uppers which I can stack at less than 2" thick. Also very light. My Merrell Trail Gloves weight about 7oz each / 14oz a pair. Vapor Gloves are 5.5 oz each. Would take some getting used to for running and don't want to do too much too fast. Other options are Vivobarefoot, Altra (cushioned soles, but zero drop - heel thickness same as forefoot, unlike most shoes with thicker heels), New Balance Minimus.
To go whole hog on minimal, Vibram Five Fingers. Also good for gym, imo.