How mirror of quintic was originally found?

The history of this is as follows. In the paper by Candelas, Lynker and Schimmrigk there are two weighted hypersurfaces whose cohomology is mirror to that of the quintic. These therefore are two potential candidates for the mirror quintic. The question then was how to decide whether they provide mirror partners to the quintic or not. This was addressed in a paper by Lynker and Schimmrigk (http://inspirehep.net/record/27957) by transporting the Greene-Plesser construction of quotients of conformal field theories to the level of Landau-Ginzburg theories and hence weighted hypersurfaces. This established at level of physics that the two weighted hypersurfaces in the list of Candelas-Lynker-Schimmrigk are isomorphic and that at the level of physics they are both mirrors of the 1-parameter quintic family.

The account Brian Greene gives in The Elegant Universe (a surprisingly useful reference for the early history of mirror symmetry) is that he and Plesser were studying techniques to create new Calabi-Yau manifolds using orbifolds. The mirror of the Fermat quintic arises naturally in this context. Nearly simultaneously, Candelas, Lynker, and Schimmrigk generated a large number of examples of probable mirror pairs by studying Calabi-Yau hypersurfaces in weighted projective spaces. (Math Reviews was highly enthusiastic: http://www.ams.org/mathscinet-getitem?mr=1067295 ).

The history is described here: The quintic mirror was constructed by Greene and Plesser as one of a few hundred mirror manifold pairs. Candelas et al. acknowledge in their article that they got the quintic mirror from Greene and Plesser.

There is an interesting quote by Brian Greene why he did not himself pursue the enumeration problem solved by Candelas et al. using the quintic duality that he and Plesser had discovered:

You can have an equation that you know is abstractly correct, but it can nevertheless be a major challenge to evaluate that equation with adequate precision to extract numbers from it. We had the equation but we didn't have the tools to leverage it into the determination of numbers. Candelas and his collaborators developed the tools to do that, which was a huge accomplishment.

The Greene and Plesser construction as it is summarized here: