उदकस्योदः संज्ञायाम्

Operational rule vidhi

उद is substituted for उदक when the compound is a Name.

Word by word (padaccheda)

Target उदकस्य ṣaṣṭhī · ekavacanaSubstitute / term उदः prathamā · ekavacanaRight context संज्ञायाम् saptamī · ekavacana

Target Ṣaṣṭhī (6th case): names what gets replaced, "in place of X" (1.1.49 षष्ठी स्थानेयोगा).
Substitute / term Prathamā (1st case): in operational rules, the substitute or the thing introduced; in definitions, the term or its referent.
Right context Saptamī (7th case): "when X follows", the operation applies to what precedes (1.1.66 तस्मिन्निति निर्दिष्टे पूर्वस्य). Also a condition or domain ("in the sense of").

Full reading

उदकस्य उदः संज्ञायाम्

Pāṇini leaves out words that carry over from earlier sūtras. Expand to see the rule as a reader actually understands it.

As a rewrite rule (sketch from case endings)

उदकस्य → उदः / संज्ञायाम्

Replace udakasya with udaḥ when saṃjñāyām follows. Inherited words (anuvṛtti) may add further conditions.

Commentary

उद is substituted for उदक when the compound is a Name.
Thus उदमेघः 'a person called Udamegha', उदवाहः "a person named Udavahah." The well-known Patronymics from these are औदमेघिः and औदवाहिः ॥ Why do we say 'when it is a Name'? Observe उदकगिरिः ॥

Vart:- उद is the substitute for उदक when it stands as the second member of a compound and denotes a Name: as, लोहितोदः, नीलोदः, क्षीरोदः ॥

Śrīśa Chandra Vasu, The Ashtadhyayi of Panini (1897), via ashtadhyayi.com.