In strong shape theory, you remember all of the homotopy coherence information in the approximations to a topological space.
In classical shape theory or Čech homotopy, the polyhedra or simplicial sets approximating a given space are treated as giving a pro-object in the homotopy category. In the early strong versions of shape theory the diagrams are thought of as being in the coherent homotopy pro category of simplical sets, or one of the equivalent forms (e.g. that due to Edwards and Hastings (1973)). This is now probably better treated by the methods discussed in the entry on shape theory.
The earliest paper on strong shape would seem to be by D.E. Christie, in 1944. He looked at a 2-truncated version of the later theory. The next appearance of the idea would seem to be in the 1970s when more or less independently, Edwards and Hastings, Porter, Mardesic and Quigley came out with various approaches to the problem of structuring shape theory in a richer manner. The availability of work on homotopy coherence, homotopy limits and colimits, etc. would seem to have been one of the key steps n this development.
D.A. Edwards and H. M. Hastings, (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag, pdf
J.T. Lisica and S. Mardešić, Coherent prohomotopy and strong shape theory, Glasnik Mat. 19(39) (1984) 335–399.
S. Mardešić, Strong shape and homology, Springer monographs in mathematics, Springer-Verlag.
Michael Batanin, Categorical strong shape theory, Cahiers Topologie Géom. Différentielle Catég. 38 (1997), no. 1, 3–66.
T. Porter, Stability Results for Topological Spaces, Math. Zeit. 150, 1974, pp. 1-21.
T. Porter, Abstract homotopy theory in procategories, Cahiers Top. Géom. Diff., 17, 1976, pp. 113-124, numdam
T. Porter, Coherent prohomotopical algebra, numdam, Cahiers Top. Géom. Diff. 18, (1978) pp. 139-179;
T. Porter, Coherent prohomotopy theory, Cahiers Top. Géom. Diff. 19, (1978) pp. 3-46, numdam
F. Cathey, Jack Segal, Strong shape theory and resolutions, Topology and its Appl. 15 (1983) 119–130
Bernd Günther, Strong shape of compact Hausdorff spaces, Topology and its Applications 42:2, 1991, pp. 165–174, doi; A Tom Dieck theorem for strong shape theory, Trans. Amer. Math. Soc. 338 (1993), 857–870 doi; The use of semisimplicial complexes in strong shape theory, Glas. Mat. Ser. III 27 (47) 1992, no. 1, 101–144 (contains quasicategory ideas in strong shape) gBooks ; Comparison of the coherent pro-homotopy theories of Edwards-Hastings, Lisica-Mardešić and Günther, Glas. Mat. Ser. III 26(46) (1991), no. 1-2, 141–176