What is the son of Varuna in Hindu mythology?

 Varuna has many sons mentioned in various Hindu scriptures, from the Vedas to the Ramayana and Mahabharata to the Puranas:

• Pushkara the Nourisher, who married Soma’s daughter Jyotsnakali.

• Bala, a Deva of strength.
• Vasishtha, a famous Vedic sage.

• Agastya, another famous Vedic sage.

• Bhrigu, another famous Vedic sage.

• Vaidya, who still resides with his father in Varuna Loka and helps safeguard it with his powers. He also serves under Lord Indra as one of the Rain Devas, and is worshipped for rain, for relief from diseases (especially colds, coughs, and diabetes), for protection of fishers and sea travelers, and various other purposes. The Puranas actually have rather a lot to say about this particular son of Varuna.

• Valmiki

• Vandi, a Saint who presided over an assembly of philosophers in the court of Raja Janaka.
• Sushena, a vanara who helped Lord Rama.
• Hiranya
• Harita
• Jalandhara
• Srutayudha, a heroic king who fought in the Kurukshetra war on the side of the Kauravas. He unwisely hurled the supernatural mace of his father Varuna at the unarmed charioteer Shri Krishna. Rebelling against this adharmic act, the mace returned the stroke upon Srutayudha and killed him.

• Some say the demon Kali is a son of Varuna (no relation to the Goddess).
• Adharmaka - though this may just be another name of Kali.
• It is prophesied in Shrimad Bhagavata Mahapurana 3.24 that Varuna will have another son in the future: Daksha Savarni Manu, who will fulfill the role of Manu in a future age to come.

Varuna also has many daughters but since this question only asks about his son(s) I will leave it at that.

Who is the Varuna God?

 Rig Veda is a spiritual treatise. Rishis of Rig Veda composed Hymns to communicate to the World their spiritual experiences in esoteric sense.

The basic premise of Spirituality is that every Human has within oneself a DIVINITY, concealed by layers of ignorance or evil thinking/habits.

These layers of ignorance or evil thinking/habits, which are deep rooted, will keep on creating obstacles or divert the attention of the spiritual practitioner from realising the concealed DIVINITY within oneself, unless the DIVINE help/assistance is assured.

According to Rig Veda, Varuņa, the son of Aditi (mother of Gods), is the divinity which provides the required assistance to humans in overcoming those layers of ignorance or evil thinking/habits.

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Rishis of Rig Veda termed, in esoteric sense, those layers of ignorance or evil thinking/habits as Pāśa (पाश) indicating a "noose," "rope," "snare," or "fetter" and the obstacles created as Vṛtrá/Paṇi/Vala/Śambara/Bṛsaya.

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The seer śunaḥśepa explicitly requests Varuņa to release or loosen the three knots or bonds.

Rig Veda 1.24.13

शुनःशेपो ह्यह्वद्गृभीतस्त्रिष्वादित्यं द्रुपदेषु बद्धः । अवैनं राजा वरुणः ससृज्याद्विद्वाँ अदब्धो वि मुमोक्तु पाशान् ॥
śunaḥśepo hy ahvad gṛbhītas triṣv ādityaṃ drupadeṣu baddhaḥ | avainaṃ rājā varuṇaḥ sasṛjyād vidvām̐ adabdho vi mumoktu pāśān ||

English translation:

Since Śunaḥśepa, seized and bound in three stocks, called upon the Āditya, King Varuṇa should set him loose. Let him—the knowing one, never cheated—release the fetters ( mumoktu pāśān).

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The three bonds/fetters, by which King Varuṇa binds an individual from not progressing in spiritual path are:

1. anṛta (अनृत) - not abiding by the path of TRUTH, with
2. steya (स्तेय ) - committing of any theft or misdeed
3. parigraha (परिग्रह) - possessiveness, greed, and attachment to wealth, status, or items beyond one's needs

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Thus, an individual is expected to overcome the above 3 vices, and habituate oneself to the following 3 positive traits. King Varuṇa will assist in progress.

1. Follow the path of ऋतम् - ṛtam or the DIVINE Law/TRUTH , with trikāraṇaśuddhi - त्रिकारणशुद्धि (purity in thoughts, words and action).
2. अस्तेय -asteya - non-committing of any theft or misdeed with trikāraṇaśuddhi
3. Aparigraha (अपरिग्रह) - non-possessiveness (not accumulating/keeping more than required) with trikāraṇaśuddhi

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Rig Veda says, when an individual obeys the commandments of King Varuṇa, one will be bestowed with blessings what one cannot think of. That means one will reach spiritual heights.

Rig Veda 7.60.6

इमे मित्रो वरुणो दूळभासोऽचेतसं चिच्चितयन्ति दक्षैः । अपि क्रतुं सुचेतसं वतन्तस्तिरश्चिदंहः सुपथा नयन्ति ॥
ime mitro varuṇo dūḻabhāso 'cetasaṃ cic citayanti dakṣaiḥ | api kratuṃ sucetasaṃ vatantas tiraś cid aṃhaḥ supathā nayanti ||

English translation:

These hard to deceive—Mitra, Varuṇa, (and Aryaman)—cause even the unperceptive man to perceive through their skills. Fully knowing the resolve based on good perception, they lead by a good path even across narrow straits.