It is not named after a person, but I would argue that the inequality does have a name, "the convexity inequality for xp". If you look at the definition of a convex function, the statement that f(x) = xp is convex is exactly the statement that ((x+y)/2)p ≤ (xp+yp)/2. (Plus either that f is continuous or that the weighted version also holds). It is also correct to say "is a consequence of convexity", but it seems better to call it the statement of convexity than a consequence of convexity.
The multivariate form of the convexity inequality is named after a person; it is Jensen's inequality.
All of this may seem like a pat answer, but it works. It is a theorem that you get a valid norm if xp is replaced by any non-negative convex function φ(x) with suitable behavior at 0 and ∞. The resulting norm is called an Orlicz norm and the resulting Banach space is called an Orlicz space.