When it applies to a person's name, according to Vedic philosophy it means identifying oneself with the universe or ultimate reality. Some say that when a child is born it cries Koham-Koham which means Who am I? That is when the universe replies back Soham. You are the same as I am. It also stems from the Sanskrit word which means, "self pride."
When used for meditation, "Soham" acts as a naturalmantra to control one's breathing pattern, to help achieve deep breath, and to gain concentration.
Sooooo... is the sound of inhalation, and is remembered in the mind along with that inhalation.
Hummmm... is the sound of exhalation, and is remembered in the mind along with that exhalation.
Soham is also considered a mantra in Tantrism and Kriya Yoga, known also as Ajapa mantra, Ajapa Gayatri, Hamsa Gayatri, Hamsa mantra, prana mantra, Shri Paraprasada mantra, paramatma-mantra, and as such used notably on its own, in the meditation practice ajapa japa and in the kriya practice shabda sanchalana.
The mantra is also inverted from so 'ham (the sandhi of saḥ + aham) to ham + sa. The combination of so 'haṃ haṃsaḥ has also been interpreted as "I myself am the Swan", where the swan symbolizes the Atman. An etymology of haṃsa "swan, goose" (in fact cognate with English goose) as from ahaṃ sa "I am that" is found in the 14th century commentary on the Vedas by Sayana (14th century).
so 'ham is an emphatic form of aham, the first-person pronoun ("I"), translating to "I myself". Interpreted as a nominal sentence, it can also be read as "It/He is I". Its use as a mantra emerges in mystical Sanskrit literature of the medieval period. The mantra is sometimes claimed to originate with the Isha Upanishad (verse 16), which ends:
yat te rūpaṃ kalyāṇatamaṃ tat te paśyāmi yo 'sāv [asau puruṣaḥ] so 'ham asmi
"The light which is thy fairest form, I see it. I am [that] what He is (viz. the person in the sun)" (trans. Max Müller)
Swami Muktananda - although teaching the traditional So on inhalation and ham on exhalation as a letter from 1968 to Franklin Jones reveals - later published a book teaching Ham on inhalation and sa on exhalation. This practice is described in several later books all referring to Muktananda.
However, this verse 155a is not found in the Vijnana Bhairava first published in 1918 in the Kashmir Series of Text and Studies but is quoted from a commentary by the Abhinavagupta disciple Kṣemarāja in his Shiva Sutra Vimarshini (commentary on the Shiva Sutras) in later editions of Vijnana Bhairava.
^In English language literature also printed as So’ham, So Ham, So-aham, Sohum, So Hum, Saham, Sa'ham, Sau-ha, Sah-karena/Sahkara=the sound of Sa
^Satyananda Saraswati (1989). Yoga and Kriya: A Systematic Course in the Ancient Tantric Techniques of (2 ed.). Munger, Bihar, India: Bihar School of Yoga. p. 497 ff. ISBN978-81-85787-08-4. "Listen carefully to your breath; you will hear the sound So with inhalation and Ham with exhalation."
^Satyananda Saraswati (1989). Yoga and Kriya: A Systematic Course in the Ancient Tantric Techniques of (2 ed.). Munger, Bihar, India: Bihar School of Yoga. p. 668 ff. ISBN978-81-85787-08-4.
^"Yoga Chudamani Upanishad". Retrieved 2009-05-17. "This mantra which is called “Ajapa Gayatri” (…) goes outside with the sound “ha” and goes again inside with the sound “sa”."
^"Yoga Sikha Upanishad". Retrieved 2009-05-17. "6.53 The prana goes out with sound “ham” and goes in with the word “sa”, and all beings naturally chant the mantra “Hamsa, Hamsa” (while exhaling and inhaling)."
^Renfrew Brooks, Douglas (2000). Meditation revolution: a history and theology of the Siddha Yoga lineage. Motilal Banarsidass Publ. p. 510. ISBN978-81-208-1648-0.
^Avalon, Arthur (2004). Mahanirvana Tantra Of The Great Liberation. Kessinger Publishing. p. 82. ISBN978-1-4191-3207-0. "All beings say the ajapa Gayatri, which is the expulsion of the breath by Hangkara, and its inspiration by Shakara"
^Dayal, P (1991). Raja Rao : A Study of His Novels. Atlantic Publishers & Distributors. p. 53. ISBN81-7156-160-8. "The Mahanirvana Tantra unequivocally specifies an identity between jiva and Brahman (...) The idea of "So’ham" (I am He or I am one with the Supreme) is explicitly emphasized in this Tantric text."
^Mahanirvana Tantra is claimed to be a juridical fabrication in: Duncan, John (1978). Essays in classical and modern Hindu law. BRILL. p. 197 ff. ISBN978-90-04-04808-9.
^"The Yogavishaya of Minanath". Retrieved 2009-05-17. "Hamsa Hamsa is the mantra upon which depends the bodies of living creates. It is meditated on as the collective form of vital breath in the knots.  21600 times daily the word Hamsa is being pronounced -- in this way one constantly meditates 'So-aham'. "
^Feuerstein, Georg (2002). The Yoga Tradition: Its History, Literature, Philosophy and Practice. Motilal Banarsidass Publ. p. 537. ISBN978-81-208-1923-8. "[The psyche] exits [the body] with the sound ha and reenters with the sound sa."
^Siddha Guru Gorakhnath. Brahmamitra Awasthi, ed. Yoga Bija. Delhi, India: Swami Keshwananda Yoga Institute. p. 112.
^"Goraksha Shataka v42". Retrieved 2009-05-17. "42. With the sound of 'ha' jiva (in the form of prana) goes out; with the sound of 'sa' (in the form of apana) it enters (the body) again. The jiva repeats continually that mantra 'hamsa, hamsa'."
^Olson, Carl (2007). Hindu primary sources: a sectarian reader. Rutgers University Press. p. 439. ISBN978-0-8135-4070-2.
^Kiehnle, Catharina (1997). Songs on yoga: texts and teachings of the Mahārāṣṭrian Nāths. Franz Steiner Verlag. pp. 155–56. ISBN978-3-515-06922-9.
^"Shiva Svarodaya (51)". Retrieved 2009-05-17. "The Shiva Svarodaya scripture's verse 51 says, "The process of exhalation is said to contain the letter ham, and the inhalation contains the letter sa.""
^Mallinson, James (2004). Gheranda samhita: the original Sanskrit and an English translation. YogaVidya.com. p. 127. ISBN978-0-9716466-3-6.
^Ma Yoga Shakti (1995). Gheranda samhita. La scienza dello yoga. Edizioni Studio Tesi. p. 181. ISBN978-88-272-1099-4.
^Yogi Pranavananda (2000). Tony Rodriguez, ed. Pure Yoga. Motilal Banarsidass Publ. p. 113 ff. ISBN978-81-208-1508-7. "With the sound 'Sah' the breath goes in; with the sound 'Ham' the breath comes out"
^"Gheranda Samhita 5:84". Retrieved 2009-05-17. "Gheranda Samhita 5:84 indicates, "Breath of every person, in entering, makes the sound of 'sa', and in coming out (bahiryati), that of 'ham.' ""
^Singh, Panchanan (2004). The Forceful Yoga: Being the Translation of Haṭhayoga-pradīpikā, Gheraṇḍa-saṃhitā, and Śiva-saṃhitā. Motilal Banarsidass Publ. p. 275. ISBN978-81-208-2055-5.
^Satyananda Saraswati (1989). Yoga and Kriya: A Systematic Course in the Ancient Tantric Techniques of (2 ed.). Munger, Bihar, India: Bihar School of Yoga. p. 497 ff. ISBN978-81-85787-08-4.
^Devanand, G. K. Teaching of Yoga. Himalayan Institute Press. p. 116. ISBN978-81-313-0172-2. "Soham is a universal mantra vibration, with Sooo.... being remembered with inhalation and Hummm... being remembered with exhalation."
^Mumford, John (1999). Death: beginning or end? : methods for immortality. Llewellyn Worldwide. p. 97 ff. ISBN978-1-56718-476-1.
^Stutley, Margaret and James (1977). A dictionary of Hinduism : its mythology, folklore, and development 1500 B.C.-A.D. 1500. London: Routledge. p. 372. ISBN0-7100-8398-X. "The Hamsa symbolizes knowledge and the life-force or cosmic breath (prana), 'ham' being its exhalation, and 'sa', its inhalation which is regarded as the return of the individual life-force to brahman, its cosmic source."
^Srinivasan, N. K. (2007). Safe and Simple Steps to Fruitful Meditation. Pustak Mahal. pp. 48–49. ISBN978-81-223-0891-4.
^Tigunait, Pandit Rajmani (2000). Power of Mantra and the Mystery of Initiation. Himalayan Institute Press. p. 68 ff. ISBN978-0-89389-176-3. "(..) you will hear the sound sooo in the inhalation and hammmmm in the exhalation."
^"Letter from Swami Muktananda to Franklin Jones, April 23, 1968". Retrieved 2009-05-17. "Harmonize the repetition of mantra with the breathing as follows: With "So" take it in and with "ham" bring it out. (...) When one's mind is fixed on "So" with the incoming breath and on "ham" with the outgoing breath it is mantra-japa. (...) Your beauty, your energy, your duty, your religion, your Guru and guide; your study, worship and prayer -- all lie in engaging yourself to the remembrance and repetition of "So'ham", "So'ham". This is my instruction, this is my precept. This is to followed or practiced, and reflected upon devoutly."[dead link]
^Swami Muktananda (1992). I Am that: The Science of Hamsa from the Vijnana Bhairava. SYDA Foundation. p. 27 ff. ISBN978-0-914602-27-9. "Sit quietly, and watch the going out and coming in of the breath . . . Bhairava says that as the breath comes in, it makes the sound ham, and as the breath goes out, it makes the sound sa. This is known as ajapa-japa, the unrepeated mantra repetition. One who simply watches the breath, being aware that it is coming in and going out with the sounds ham and sa, is doing ajapa-japa, and this is the true way of practicing mantra."
^Renfrew Brooks, Douglas (2000). Meditation revolution: a history and theology of the Siddha Yoga lineage. Motilal Banarsidass Publ. p. 509 ff. ISBN978-81-208-1648-0.
^Singh, Jaideva (1991). The Yoga of delight, wonder, and astonishment: a translation of the Vijñāna-bhairava. SUNY Press. p. 143 ff. ISBN978-0-7914-1073-8. reprinted and published as: Singh, Jaideva (2002). Vijnanabhairava or Divine Consciousness: A Treasury of 112 Types of Yoga. Motilal Banarsidass Publ. p. 173. ISBN978-81-208-0820-1.