अवृद्धाभ्यो नदीमानुषीभ्यस्तन्नामिकाभ्यः

The affix अण् comes in the sense of descendant after the words are the names of rivers or women when such words are not वृद्धम् words and when they are used as names and not as adjectives.
The word Vriddham in the sutra, does not mean the technical Patronymic of that name as defined in other Grammars and used in Sutras like 4.1.166, but means the Vriddham as defined in this Grammar i. e. a word whose first syllable is a Vriddhi letter 1.1.73. The word अवृद्धाभ्यः shows the orthography of the word to which the affix is to be added. The word नदीमानुषीभ्यः declares the meaning of that word and both attributes refer to one and the same base, i.e. orthographically the base should have a Vriddhi letter in the first syllable, and etymologically it should denote a man or a river. The word तन्नामिकाभ्यः applies both to the base and the affix.

Thus यमुना--यामुनः 'a son of the Yamuna.' ऐरावत 'son of the Iravati'. वैतस्तः 'son of the Vitasta.' नार्मदः 'son of the Narmadda.' These are all names of rivers. Similarly शैक्षितः 'son of Sikshita' चैन्तित 'son of चिन्तिता' ॥

Why do we say 'when not a Vriddha word'? Observe चान्द्रभाग्याया अपत्यं = चान्द्रभागेयः; so also वासवदत्तेयः formed by ढक् 4.1.120. Why do we say being 'names of rivers or women.' Observe सौपर्णेयः and वैनतेयः ॥ Why do we say "when it is a name?" Observe शोभनाया अपत्यं = शौभनेयः 'the son of a beautiful woman.'

ढक् 4.1.120 is the general affix by which patronymics are formed from feminine bases. The present sutra is an exception to that.