The Devi Bhagavata gives a bunch more, although some only seem to appear in the Hari Vamsha.
Vyāsa said :-- The part incarnations of Suras and Asuras on this earth, and their names I am now saying to you in brief; hear.
Vāsudeva: Kaśyapa (by the way this probably means Kaśyapa is an incarnation of the first human as Vasudeva is also called an incarnation of him)(this implies the personal name of the first human is Ariṣṭanemi)
Vāsudeva Śrī Kṛṣṇa: Śrīmān Nārāyaṇa; the son of Dharma
Arjuṇa: of Nara
Yuidhiṣṭhira: part incarnate of Dharma
Bhimasena: [part incarnate] of Vāyu
Nakul and Sahadeva: [part incarnate] of Aśvinī-kumāras
Karṇa, born of Kuntī: part incarnate of the Sun
Vidura: the knower of the Supreme Essence: incarnate of Yama, the king Dharmarāj.
Droṇa: the Ācārya of the Kurus and the Pāṇḍavas: part incarnate of Brihaspatī
Aśvatthāmā: part incarnate of Rudra Deva. (Later: The powerful son of Droṇa, Aśvatthāmā, though known as the part incarnate of Rudra, was really born of the four parts of Yama, Rudra, Cupidity and Anger.)
Śantanu: part incarnate of the Ocean;
his wife: of the river Ganges [Ganga] in human farm.
It is stated in the Purāṇas that the king Devaka: part incarnate of the Lord of Gandarvas
Bhīṣma Deva: the incarnate of Vasu
Virāṭa: the Lord of Matsya: the part incarnate of Marut
Dhritarāṣṭra: [part incarnate of] the Daitya Hamsa, the son of Aṛṣṭa Nemi. (He is also called the monarch of the Gandharvas in some places)
Kripa and Krita Varmā: [part incarnate of] Maruts. (This fits with the Mahabharata declaring Kripa is born of the tribe of Rudras through possession as described with Varchas later i.e. Kripa is of the tribe of the Rudras, a part incarnation of the Maruts and possesses a human/Rishi (not sure). This works because the Mahabharata says he was famous on Earth as Kripa.)
Duryodhana: [part incarnate of] Kali
Śakuni: [part incarnate of] Dvāpara;
Suvarcākhya Somapraru (???): [part incarnate of] the son of the Moon
Dhṛṣṭadyumna: [part incarnate of] part incarnate of Fire
Śikhaṇḍī: [part incarnate of] Rāksasa
Pradyumna: part incarnate of Sanatkumāra
king Drupada: was part incarnate of Varuṇa (This fits with the Mahabharata declaring Drupada is born of the Maruts through possession as described with Varchas later i.e. Drupada is a Maruta, a part incarnation of the Varuṇa and possesses a human. This works because the Mahabharata says he was famous on Earth as Drupada.)
Draupadī: [part incarnate of] Lakṣmī
Draupadī’s five sons: [part incarnate of] Visve-devas
Kuntī: was incarnate of Siddhi;
the wives of Śrī Kṛṣṇa: heavenly public women;
Thus, all the Devas came as their part incarnations, urged on by Indra.
Śiśupāla: the incarnate of Hiraṇyakaśipu
Keśī: of Haya Śirā.
The Asura named Aṛṣṭa of the form of a cow that was killed by Kṛṣṇa: son of Bali (???) (apparently his birth name in that life is Kakudmi and he is called Aṛṣṭa because people remember who he was)
Dhṛṣṭaketu: part incarnate of Anuhrādha
Bhagadatta: [part incarnate of] Vāskala
Pralamba: [part incarnate of] Lamba
Dhenuka: [part incarnate of] Khara.
Cāṇūra and Muśṭika: the two athletes: part incarnates of Vārāha, and Kiśora, the two dreadful Daityas.
Kubalaya: the elephant of Kaṃsa: was part incarnate of Aṛṣṭa, the sun of Diti (apparently he was also known as Ristha).
Vakī: the daughter of Bali (???)
Vaka: was her younger (???)(why are the children of Bali not named?)
The Daityas and Rāksasas that were born to relieve the heavy burden of the Earth: all incarnates of Asuras.
O king! I have thus narrated to you in order the incarnations of the Suras and Asuras, as they are stated duly in the Purāṇas.
While the other answers are good, I think there is a contradiction in the obvious answers to who Abhimanyu is an incarnation of. As said by Keshav Srinivasan, Arjuna seems to be the incarnation of the son of Soma, Varchas. However, he is also called an incarnation of Soma in the 15th book of the Mahabharata.
He that took his birth as the son of Arjuna, that gladdener of all, that heir to the possessions of the Pandavas, who was slain by six great car-warriors (fighting together), was Soma. He was born of Subhadra. Through Yoga-puissance he had divided
As I see it there are two ways to resolve the conflict. One Abhimanyu is Varchas and Soma for some reason calls him by a different name. The other which I think makes more sense, because it also clears up other contradictions about Abhimanyu (as discussed here) is that Varchas is one of the Asuras that controls the humans in the Mahabharata as discussed in this story here. Varchas is thus the Asura son of Soma, who is a partial incarnation of Soma.
This is actually supported strongly by examining the context of the mention of Varchas. The beginning of the page is talking about Asuras incarnating on Earth. This means expecting the reader to reach this conclusion is not a stretch. Soma also refers to him an Varchas as separate from the celestials, which makes more sense if Varchas is an Asura.
And he who was known as the mighty Varchas, the son of Soma, became Abhimanyu of wonderful deeds, the son of Arjuna. And before his incarnation, O king, the god Soma had said these words to the celestials, 'I cannot give (part with) my son. He is dearer to me than life itself. Let this be the compact and let it be not transgressed. The destruction of the Asuras on earth is the work of the celestials, and, therefore, it is our work as well. Let this Varchas, therefore, go thither, but let him not stay there long.
Thus, to recap, Soma's son Varchas, who is a partial incarnation of Soma, possesses Arjuna's son Abhimanyu and thus becomes known as Abhimanyu. He does this after Abhimanyu is born. The soul controlling Abhimanyu from birth is unknown.
Parikshit's, Yuyutsu's, Duhsala's, Sanjaya's and Janamejaya's origins are notably missing. Missing on the other side is Lavana, who while mentioned in a speech about Asuras he has to kill again by Narada, there is just a random cliffhanger about who he turns into.
Edit: (In response to an anonymous edit suggestion) I'm fine with people adding to this post, but the additions should be sourced and not in a position in which it implies they are from another text than they are.