I know that there are 12 Surya namaskar mantra that are also described in question here Correct way of Surya namaskar?. But I want to know what is the origin of Surya namskar mantra? Which scripture describes about it?
Sri Swami Satyanada in his book Surya Namaskar - A technique of Solar Visualization says that Sun has different name in each Rashi and these names are chosen for Surya Namaskara Mantras. Sri Swami Satyanada was disciple of Sri Swami Sivananda.
This is what Sri Swami Satyanada says in Chapter 6 of this book.
Every year the sun passes through twelve different phases, known as the signs of the zodiac in Western astrology, and as the rashis in Hindu astrology. According to Hindu astrology each rashi has specific attributes or moods, and in each of these twelve moods the sun is given a different name. These twelve names comprise the twelve sun mantras, which are to be mentally repeated in their respective order in conjunction with the twelve movements of surya namaskara. These sun mantras are not merely names of the sun, every sound syllable contained within them is the vehicle of a basic eternal energy (shakti) represented by the sun itself. By repetition and concentration on these mantras, the whole mental structure will be benefited and uplifted. Although these mantras do not require intellectual understanding, a translation of their meaning is given below for those with an enquiring mind, as well as for the more spiritually inclined who wish to use the mantras as a form of attunement with the source of spiritual illumination symbolized by the sun.
First of all,this is not the most precise or perfect answer.
But,the 12 Surya namsakara mantras seems to have been originated from the "Surya Mahatmya: Avatarana, Dwadasadityas, Konaarka and Worship" chapter of the Brahma Purana.
A "Surya Dwadasa Namavali"(12 names of Surya) as well as a "Surya Asthottara"(108 names of Surya) are included in this chapter.
Dwadashiva pruthaktena taani vakshaamaseshatah, Adityah Savitaa Suryo Mihirorkah Prabhaakarah/ Maartando Bhaaskaro Bhanuschitra Bhaanu Divaakarah, Ravirdwaadashabhisteshaam Jneyah Saamaanya naamabhih/ Visnurdhaataa Bhagah Pushaa Mitrendrou Varunyornamaa Vivaswaa namsumaam Twashtaa Parjanyo Dwaadashah smritaah/ Ityetey Dwadasha –adittaah pruthaktena Vyavastithaah, Uttishthanti Sadaahyetey maasthairvaa Dasabhih kramaat/
Normally Surya Deva’s names in vogue are Aditya, Savitru, Surya, Mihira, Arka, Prabhakara, Martanada, Bhaskara, Bhanu,Chitrabhanu, Divakara, Ravi and so on. But the additional names are Vishnu, Dhata, Bhaga, Pusha, Mitra, Indra, Varuna, Aryama, Vivasvata, Amsuvan, Twashta and Parjanya.
A "Surya Dwadasa nama stotram",different from the one given above, is also found in the Brahmanda Purana.
Even this page seems to confirm the claim that Brahma Purana is the origin of the 12 names/mantras.
I will try to give answer to your question about origin of Surya NamskarMantras.-
Their are 108 various names of Surya discribed in ancient scriptures eg. these names can be found in 4 vedas. i.e. Rugved (ऋग्वेद), Atherva Ved (अथर्ववेद) , Yajur Ved (यजुर्वेद) and Sam Ved (सामवेद)
Their are many Vedic refrences for origin of these 108 names , Like , Adityahridayam (आदित्य रहृदयम).
Some of the shlokas are given below -
आदित्यः सविता सूर्यः खगः पूषा गभस्तिमान्। सुवर्णसदृशो भानुर्हिरण्यरेता दिवाकरः॥ १०॥
हिरण्यगर्भः शिशिरस्तपनो भास्करो रविः अग्निगर्भोऽदितेः पुत्रः शङ्खः शिशिरनाशनः॥ १२॥
Adityahridayam (आदित्य रहृदयम).
Their are also refrences from "Surya Gayatri (सूर्य गायत्री) shlokas ,which contain some of these 108 names. Some of the shlokas are.
1) ऊँ आदित्याय विदमहे दिवाकराय धीमहि तन्न: सूर्य: प्रचोदयात
2) ऊँ भास्कराय विदमहे दिवा कराया धीमहि तन्नो सूर्य प्रचोदयात
So we can say that , the 12 surya Namskar Mantras origin is from vedas. The 12 surya Namskar Mantras are created using various names discribed in all above mentioned and various other sources. Like Surya Gayatri and Adityahridayam.
Please check the links for more info about 108 Names of Sun and Surya Gayatri and Adityahridayam.