In the cases of Germanic, Hellenic and Phrygian, the noun declines with a short *o throughout. This can be explained by the fact that, in forms of the paradigm with the sequence *h₃nóh₁m̥n- (as opposed to *h₃nóh₁mn̥), PIE sound laws would delete the laryngeal and lengthen the vowel, leading to a form *h₃nṓmn- appearing in front of endings with a vowel, and the form *h₃nóh₁mn̥ elsewhere. Presumably this would have applied at a PIE level. These particular languages would then apply Osthoff's law (or an equivalent) which would shorten the vowel when preceding a resonant closing the syllable. Thus *h₃nṓmn- > *h₃nómn-. Germanic bases its declension on the original collective form, which, taking into account a generalisation of the o-grade and the sound changes I just described, would provide this paradigm:
After which analogy could conceivably shorten the vowel in the nominative, vocative and accusative endings.
A similar sequence of events would presumably have to happen early in Greek and Phrygian's histories, which considering the ubiquitiousness of Osthoff's law in Ancient Greek and the presumed close development of the two languages, is not at all unlikely.
Osthoff's law presumably also applied in Armenian and Italic, leading to a similar alternation between long and short o-grades throughout the paradigm. Italic appears to have generalised the long *o from the nomino-accusative, although influence from words like cognōmen (connected with the IE root ǵneh₃-) cannot be discounted. Armenian could have conceivably done the same thing, but it is also possible that the forms simply merged through phonological change, as below:
