That is at best a convention. You can multiply the Lagrangian by -1, without changing the equations of motion or the resulting trajectories. So physically nothing changes, but maxima and minima swap.

You need a convex Lagrangian for the Legendre transform to work so I think it's more than a convention. Of course you could redefine the Legendre transform to work for concave functions instead of convex functions but in either case you'll want the convexity or concavity of Lagrangians to be uniform.