A historical reason for the importance of things Lagrangian in a TQFTy setting comes from Floer Homology and a conjecture of Atiyah that ``the two'' Floer theories are the same for 3-manifolds. Floer defined one homology (monopole Floer homology) based on the Chern Simons action functional, with the generators of its chain complex (the analogues of critical points in Morse Theory) being flat connections. He built another homology theory (the more standard one, today) to deal with the Arnol'd conjecture in Hamiltonian mechanics. If you take a Heegegard splitting of the 3-manifold, then, associated to the dividing surface you have the moduli space of flat connections, which is a symplectic manifold. You can then start to set up a Hamiltian Floer theory on this moduli space. Those flat connections over the surface which extend into the three-manifold on one side of the surface define a Lagrangian submanifold of the moduli space. Those flat connections which extend to the other side define another Lagrangian submanifold. The symplectic Floer business on the moduli space, set up properly, gives you a count the intersections of these two Lagrangian submanifolds, and hence of the set flat connections which extend to the whole 3-manifold. Atiyah conjectured these two Floer theories are the same. I suggest as an entry the Math Reviews article, See MR1283871, by Donaldson on the proof of a special case of this conjecture by Deitmar Salamon and Stamatis Dostoglou.
A philosophical reason is Weinstein's "symplectic creed'creed": everything is Lagrangian. The classical version of a quantum state is a Lagrangian submanifold. So if one's TQFT yields a semi-classical' "semi-classical" Hilbert space, say a symplectic vector space, then the states (wavefunctions) of the ``Hilbert space'' "Hilbert space" be Lagrangian subspaces