Jäschke subdivides the meaning of mod pa as (a) “an argumentative of yin,” and hence “an emphatic word for to be” and “indeed, to be sure,” and (b) “an argumentative of yod, signifying abundance, plenty.” I personally do not have the feeling for the second usage unless I envision it to be mang mod kyi (or something similar). It is perhaps felicitous to consider the following structure: verb (stem) + mod (with or without kyi/kyang). The following three types are possible: yin mod (kyi/kyang), yod mod (kyi/kyang), and bshad mod (kyi/kyang).
The question is what the etymology of mod may be. Jäschke wonders (i.e. with a question mark) whether mod pa could be a cognate to mad pa (“true”). dGe-chos, on the other hand, has stated that mod in yin mod kyang means mol (“to speak” or “to speak”) and zer (“claim” or “maintain”). See the dGe chos gsung ’bum (vol. 5, p. 76).