Where Tara comes from is not settled, and the popular account of it is usually backwards.
The dating evidence. The earliest textual reference is in the Manjusri-mula-kalpa, placed somewhere between the fifth and eighth centuries, which describes her as the compassion of Avalokitesvara. The earliest images are earlier than most of the texts: Kanheri cave 90 in the sixth century and Ellora cave 6 around the seventh. Her worship was established at Nalanda by the sixth century. So a cult existed in India by the sixth century and had reached a major monastic university.
The contested part. How she relates to earlier Indian goddesses is open. Mallar Ghosh proposed that she originated as a form of Durga. The general scholarly position is more cautious: her origin is unclear and remains a subject of inquiry. There is not even agreement on what to call her, and she is variously classified as a deity, a buddha and a bodhisattva. Miranda Shaw, in Buddhist Goddesses of India of 2006, argues that motherhood is central to the conception of Tara and reads her as reflecting Mahayana's openness to women practitioners in sixth-century India.
The borrowing that runs the other way. A Hindu Tara exists among the ten Mahavidyas of Shakta tantra, and the obvious inference, that the Buddhists took her from Hinduism as they took Mahakala and Hayagriva, is the wrong one in this case. The Hindu Ugra Tara or Mahacinakrama Tara is generally held to have been imported into Hindu tantra from Buddhist sources, and her very name, referring to the practice of Great China, points at where she was understood to have come from. Direction of borrowing has to be established case by case and cannot be inferred from which religion is larger.
There is also a persistent claim linking Tara to pre-Buddhist or Bon goddesses in Tibet. It is not established, and the atlas does not repeat it as fact.