"Lost" isn't an accurate description - it's better asked as "Why did Papua New Guinea become independent?". Papua New Guinea did not begin as a unified territory. The Netherlands had claimed the western half of the island of New Guinea in 1828 (its status as a former Dutch colony was the justification for Indonesia's takeover in 1963). The next European claim was the 1883 annexation of New Ireland (New Britain and New Ireland are islands the north of the eastern tip of New Guinea). In response, Queensland annexed the SE quarter of the island of New Guinea in 1883 (Queensland is now a state in Australia, but was then, like the other Australian states, a separate colony). The UK government refused to ratify the annexation by Queensland. However, in 1884, Germany established a protectorate over the NE quarter and the Bismarck Archipelago (which includes New Britain - which became New Pomerania while under German rule - and New Ireland): German New Guinea (Deutsch-Neuguinea). In response, the British established a protectorate over the SE quarter (which had been annexed by Queensland), and then annexed this territory in 1888: Papua. After the unification of the Australian colonies in the Commonwealth of Australia in 1901, the UK handed Papua over to Australia (in 1902).

WW1 saw a prompt Australian invasion and takeover of German New Guinea, and the Paris Peace Conference of 1919 saw Australian prime minister Billy Hughes vocal in his demands for Australian possession of German New Guinea. He got his way, in the form of a League of Nations Mandate. The League of Nations mandates were intended as short-to-intermediate term control, preparing the territories for self-government and independence. Very little happened in the way of such preparation before WW2. WW2 saw the Japanese invade New Guinea, and Japan controlled New Britain to the end of the war. After the war, the Territory of New Guinea (former German New Guinea) became a United Nations Trust Territory, again with the intent of being prepared for self-government and independence. When Australia regained full control of the Territory of New Guinea, it was administered with the Territory of Papua as a single unit for the first time. This shared administration was further reflected in the legal merger of the two territories into the Territory of Papua and New Guinea in 1949.

This time, there were real steps towards self-government and independence. In 1951, the task of administration was assumed by the Legislative Council of Papua and New Guinea, which initially had 28 members, of who 3 were elected, and 3 were indigenous. For most of its existence, the Council was thoroughly dominated by appointed Australian members. The international community was not happy with the glacially slow pace towards meaningful self-government as the rest of SE Asia and Oceania was de-colonising, and this put pressure on Australia. Australia reacted by implementing reforms, and in 1960, the Council, now with 37 members, included 12 elected members, and at least 5 members had to be indigenous. This wasn't enough of a reform for the international community, and the United Nations Trusteeship Council recommended establishing a real parliament. Again, Australia did so: the House of Assembly of Papua and New Guinea first sat in 1964.

The next step was the last: independence. Papua New Guinea became independent in 1975. The House of Assembly was renamed the National Parliament of Papua New Guinea.

Why independence? Australia was responsive to international pressure to decolonise, and the Australian public was happy for Papua New Guinea to become independent.