By a theorem of Mazur-Ogus (Katz' conjecture) the $m$-dimensional Newton polygon of a variety lies above or is equal to its $m$-dimensional Hodge polygon.
A variety is ordinary if these polygons are equal (for all $m$).
For abelian varieties the $m=1$ case suffices and you see that an abelian variety is ordinary iff it is ordinary in the usual sense.
By a theorem of Grothendieck-Katz most varieties are ordinary. This is stated more precisely also in Illusie's paper you mention.
Have a look at Mazur's beautiful paper on Katz' conjecture.
Let me also talk about "constructing" varieties with given Newton polygon.
If you stick to the case of curves there are many open questions (to my knowledge). For example, Mazur asks in loc. cit. (page 659) if all five different possible Newton polygons arising from a smooth projective curve of genus $3$ allowed by the restraint of Poincaré duality really arise from some curve or not. I don't know if this question has been answered by now.
It is interesting to look at "strata" in certain moduli spaces of abelian varieties. For example, every "symmetric" Newton polygon arises from an abelian variety (and the Newton polygon of an abelian variety is symmetric). See http://arxiv.org/abs/math/0007201 .
For "strata" in Shimura varieties see
http://arxiv.org/abs/1011.3230 (Wedhorn-Viehmann) http://arxiv.org/abs/1111.6830 (Kret)
These have to do with showing existence of abelian varieties with certain Newton polygons.
