Arya was fortunate enough to train with Jaqen H'ghar, a master of the enigmatic and skilled assassin order. Under his guidance, she became a formidable assassin and infiltrator, able to change her face at will. So, why did he release Arya before her training was complete?
Jaqen accepted Arya into the order because he recognized a unique quality or personality trait in her that made her deserving of inheriting his vast knowledge and skills. Jaqen is an exceptionally intelligent and mysterious individual, with a purpose behind every action. Even his initial encounter with Arya, while imprisoned in a cage, was premeditated.
Allowing Arya to leave the order was also part of his plan, serving a specific purpose. As a servant of the God of Death, Jaqen is naturally opposed to the Night King, who represents a perversion of death. By training and releasing Arya, Jaqen ensured that she would confront this formidable adversary on his behalf. With the Starks and Winterfell directly in the Night King's path, Arya's personal stake in the battle would motivate her to surpass the efforts of any other agent from the order.