The website is clearly supported by lots of money which doesn't guarantee that it reflects the most accurate scientific information. The very page you quoted says under the picture:
The material is graphene, also known as graphite...
Well, no. Graphene is not the same thing as graphite. Graphite is a 3-dimensional material used to produce pencils - and graphene is just one layer of graphite.
So you can't expect everything to be "quite right". The honeycomb lattice is useful to get a "denser packing" which, under some circumstances, could mean a higher density of the molecules that may transform light to electromagnetic energy.
However, it is very subtle to compare "different lattices" of the "same material" because the same material usually can't form "all lattices". So I suspect that the statement is just phenomenological in character - some of the most efficient solar cells that people have found had a honeycomb lattice but as far as I know, there's no "universal proof" that it has to be so.
The situation would be much less ambiguous if one asked a well-defined geometric question - what is the best close packing of spheres, for example. However, the efficiency of solar cells also depends on more complicated physical properties of the material than just its lattice, so one shouldn't expect universal yet simple answers.