The term "Combinatorial homotopy" was coined by J.H.C. Whitehead in its use for the first two of his papers
"Combinatorial homotopy. I." Bull. Amer. Math. Soc. 55 (1949) 213--245.
"Combinatorial homotopy. II." Bull. Amer. Math. Soc. 55 (1949) 453--496.
"Simple homotopy types." Amer. J. Math. 72 (1950) 1--57.
These were rewrites and extensions of some of his very original papers which appeared in 1939-1941, in which he showed how some possible extensions of basic techniques in what became called "Combinatorial group theory", such as Tietze transformations, were in fact not generally possible, and that there were obstructions lying in what is now called the Whitehead group. So I'd be interested to know of a contrary view, but it seems to me that the impetus for the name came from combinatorial group theory.
The first paper listed covers what is now a standard basis for homotopy theory (CW-complexes, the Whitehead Theorem on homotopy equivalences, ...) and the 3rd has been very influential. The second is not so well reported on, but is one of the bases of the use of higher homotopy groupoids in homotopy theory, which I have reported in a web page under the term "Higher dimensional group theory", and which kind of extends aspects of Whitehead's approach. In particular, the free crossed modules which appeared in the second paper were a key to progress, and have a strong link with group theory and "Identities among relations".
One point which struck me recently was that you really need this concept of free crossed modules if you want to write for the boundary of the standard square diagram $\sigma$ giving the Klein bottle a non commutative formula:
$$\partial(\sigma) = a+b-a +b $$
rather than the usual $\partial (\sigma) = 2b$, which has lost a lot of information. So Whitehead argues in paper 2 that his "homotopy systems", now called "free crossed complexes", carry more information and have better realisation properties that then the still more widely used free chain complexes with a group of operators.
Of course language and the use of terms change with the decades, but it can be useful to trace some history to its roots, and try and see to what extent the original aims have been accomplished.
A relation of the word "combinatorial" to simplicial sets, and so another line on how the word might be interpreted, was given in the paper
Kan, Daniel M. A combinatorial definition of homotopy groups. Ann. of Math. (2) 67 1958 282–312.
Nov 12, 2014 I have just come across the preprint by Vezzani on motives which uses cubical rather than simplicial methods because of some technical advantages in dealing with derived functors in an algebraic geometry setting. See also this preprint by I.Patchkoria on cubical and simplicial derived functors (published in HHA).