Consider a linear chain in simple tight binding approx. (of, say, s-type orbitals) and hopping amplitude t. Depending on the sign of t you will obtain a cosine band with either a maximum or a minimum at the Gamma point. So the sign of t has a significance for the dispersion. (This is easily checked with an algebra program for a finite chain with periodic boundary conditions. Just change the sign of the off-diagonals and check the eigenvalues of the solutions changing sign.)

If you use it as a toy model, then t is just a parameter. Otherwise you'd have to calculate the hopping terms (e.g. in some LCAO approach). For adjacent s-orbitals, t turns out to be negative. The totally symmetric eigenstate (Gamma point) then is the minimum of energy. (in principle you already find this from a 2 by two matrix already)

If you see the introduction of NNN hopping as a means to obtain a closer match between the simplest 2dim cosine band an a more realistic band dispersion, then the corresponding amplitude may be seen as a fit parameter to get the best approx. within that approximation (NN & NNN hopping).

And yes I believe to remember having heard talks (well, at least one) where NNN hopping was associated with spin frustration, but I cannot remember the context and symmetry of the lattice, unfortunately.