<p>I've seen a few print time-lapse videos lately which use gyroid infill: wavy lines, which deform across layers so that the waves end up alternating between the two axes. Other than making the time-lapse videos look much cooler, what are the benefits of this infill style compared to the more common hatching or cross-hatching?</p>

From
this reference
you can read that:
A gyroid is a naturally occurring structure which be found in
  butterfly wings and even within membranes inside cells. In 2017, MIT
  researchers discovered that when graphene was shaped into a gyroid
  structure, it had exceptional strength properties at low densities.
  They then discovered however, that the crucial aspect of this was
  actually the gyroid structure itself, and that other materials such as
  plastic could benefit from this.
It is assumed that this type of infill has better properties against failure than the normal types of infill we know.
A test conducted by an author named Martin is found
here
. He printed test specimen and subjected them to bending to test the resistance against shear stress.
From the figure can be concluded that the gyroid infill has a better resistance against bending for a lower weight.
The advantages of gyroid infill over the tested infill types are:
high shear strength, and
low weight (so less filament needed).
On top of these advantages Gyroid infill prints relatively fast with respect to some other infill types and is close to
isotropic
(i.e. uniform in all orientations), meaning that is very suitable for flexible prints.