Any idea which one is Eckenstein?

I assume the leader as I think the route is Route One, which he led on first ascent.

On John Gill's site it says "The few extant photographs of him [Eckenstein] actually climbing show him holding the rope for someone else. George Abraham recalls only one instance in all the years he climbed with Eckenstein, in which the latter lead a climb: Moss Ghyll on Scafell."
Might he not therefore be the bearded man in the middle?

Except that if you look at the Lliwedd guide history thats patently not true!

Sorry I should have had a closer look at the Gill site. If you go to http://www128.pair.com/r3d4k7/Bouldering_History1.01.html you'll see that amongst the pictures that purport to be Eckenstein there is one that is a detail of the photo that you posted - and Eckenstein is the bearded man in the middle. Does your Lliwedd guide mention who the other members of the party on this ascent were?

One must be Archer Thomson, so who is the third?

Thanks for that, in the Lliwedd guide it lists Eckenstein and Archer Thompson as the two climbers on route One, no third man. Eckenstein also did most of Route Two with Archer Thomson as leader. Horned Crag was JMAT with OE. The only three person ascent I can find is Slanting Gully Wall listed as R Langford-James, Oscar Eckstein and F Langford James , so maybe it was that?