Assume for argument's sake that the two lawns are IDENTICAL in soil condition and irrigation and care, and so on etc...
Lets just say both types of grass receive a 100% equivalent amount of care for that specific type so neither has any advantage.
It's still a bit of a coin toss then, but judging from:
Bermuda grass is one of the most sun loving warm season lawn and pasture grasses. Bermuda can be easily planted from grass seed and provides a good "medium plus" turfgrass lawn in Southern areas. Bermuda turf requires a moderate amount of lawn grass care, maintenance and mowing. It is considered the "South's Grass", a favorite in texture and color for all areas of usage. This perennial grass grows in tropical, sub-tropical and the transition zones. Found extensively on lawns, golf courses, sports fields and arenas, reclamation areas, parks, coastal areas and pastures. Newer, more "cold tolerant" varieties extend Bermudas planting area further North allowing a choice of a more drought resistant species in the transition zone.
Saint Augustine lawns are a popular wide bladed (coarse) lawn planted throughout many areas of the Southeastern USA. The grass is found in Mexico, Australia, and in tropical parts of Africa. It is a warm season grass that does not handle cold weather very well. The majority of this grass is planted vegetatively as seeds are not usually available commercially each season due to production difficulties. St. Augustine is a warm season, perennial grass that is widely used. A native grass of tropical origin that extends from water marshes (salty & fresh), lagoon fringes, and sandy beach ridges. Saint Augustine grass grows from the Carolinas to the Texas Gulf coast and from mid California down. This is one of the oldest coastal grasses and in its different varieties and strains it has been found from the Carolinas to Australia and New Zealand. Several coastal areas of Africa and all through the tropics St. Augustine can be found growing. Man has managed to move it inland for ground cover and pasturage. And as long as moisture and soil requirements are met it makes for beautiful lawns. It is a thick bladed grass and with good maintenance can produce a thick, beautiful dark green lawn and requires full sun to moderate shade.
If I were forced to decide for one, such as if my life depended on it, I would have to say Bermuda ...
Only because bermuda seems to tolerate a slightly cooler temp in addition to the heat, the newer strains apparently do.
But it has been awful hot of late, IF that trend reverses then Bermuda would likely win hands down.
So if your son claimed St. Augustine, way the weather has been the past 5 years or so I would offer no argument.