I added areas and updated capitals using source [13] (see Provinces of Papua New Guinea).

The 2011 census shows 87 districts in Papua New Guinea.

Change history:

Sorin Cosoveanu provided a document copied around 2003 (by his son) from a government website which has since vanished. It gives populations (as of an unspecified date), areas, capitals, and division into districts. As it turns out, the Pacific Island Travel  website (retrieved 2010-01-13) has identical information. There are links to pages for each province. The total population is close to the 1990 census total. We have to assume that the list of districts is current as of roughly 1990-1993. Here is the list of 104 districts derived from that source.

I haven't been able to find definite information about district changes between 1990 and the present. Some changes can be inferred by comparing the 1990 list with the 2010 list, with the help of a map. These are my inferences, province by province.

1. Bougainville: apparently there were only renamings: Buin became South Bougainville, Buka became North Bougainville, and Kieta became Central Bougainville.
2. Central: Hiri and Kairuku districts merged to form Kairuku-Hiri.
3. Chimbu: Gembogl and Kundiawa districts merged to form Kundiawa-Gembogl. Names of Karimui and Sinasina changed to Karimui-Nomane and Sina Sina-Yonggomugl, respectively.
4. Eastern Highlands: The situation is unclear. Marawaka district disappeared, and Daulo, Obura-Wonenara, and Unggai-Benna districts were created. Geography suggests that Obura-Wonenara may just be a new name for Marawaka. Population suggests that something was split from Goroka, and geography suggests that it was Daulo.
5. East New Britain: Gazelle district split from something, probably Rabaul.
6. East Sepik: Wosera-Gawi and Yangoru-Saussia districts created. Its location suggests that Yangoru-Saussia was split from either Maprik or Wewak. Name of Ambunti changed to Ambunti-Dreikikir.
7. Enga: Lagaip and Porgera districts merged to form Lagaip-Porgera.
8. Gulf: Baimuru, Ihu, Kaintiba, and Malalaua districts disappeared, presumably by merging with one of the remaining districts. Geography suggests that Ihu, Kaintiba, and Malalaua merged with Kerema. Baimuru could have merged with either one. If population balance was a goal, however, Baimuru and Ihu could both have merged with Kikori.
9. Madang: Karkar and Upper Ramu districts disappeared, and Sumkar and Usino-Bundi districts were created. Since Usino and Bundi are in the upper reaches of the Ramu River, it's a good bet that Upper Ramu was renamed Usino-Bundi. It's also possible that Karkar was renamed Sumkar.
10. Milne Bay: Bolubolu, Losuia, Misima, and Rabaraba districts disappeared, and Kiriwina-Goodenough district was created. Since Losuia is on Kiriwina Island in the Trobriand group and Bolubolu is on Goodenough Island in the D'Entrecasteaux group, it's likely that Bolubolu and Losuia merged to form Kiriwina-Goodenough. Based on population and proximity, Rabaraba probably merged with Alotau. Similar factors point toward Misima merging with Samarai to form Samarai-Murua.
11. Morobe: Garaina, Kaiapit, Mumeng, and Wau districts disappeared, and Bulolo, Lae, Markham, and Nawae districts were created. Probably Lae was split from Huon. Judging by population and location, Garaina, Mumeng, and Wau may have merged to form Bulolo. Since Kaiapit is the capital of Markham, that was probably a renaming. Nawae could have been split from Finschhafen, Huon, Kaiapit, or Mumeng.
12. New Ireland: Konos, Lihir, Mussau Emira, and New Hanover districts disappeared. Based on geography, Mussau Emira and New Hanover probably merged with Kavieng. Lihir probably merged with Namatanai. Konos is about halfway between Kavieng and Namatanai, and could have merged with either one.
13. Northern: all of the districts - Afore, Ioma, Kokoda, Kira, Popondetta, and Tufi - disappeared, and Sohe and Ijivitari were created. Considering their capitals, Sohe undoubtedly was the successor to Kokoda, and Ijivitari to Popondetta. Judging by populations, the other four districts might all have merged with Sohe.
14. Sandaun: Aitape and Lumi districts merged to form Aitape-Lumi. Almost certainly Amanab and Vanimo merged to form Vanimo-Green River.
15. Southern Highlands: Ialibu and Pangia districts merged to form Ialibu-Pangia. Names of Kagua, Koroba, Mendi, Nipa, and Tari districts were extended by hyphenation. What remains to be explained is the appearance of Imbonggu and Komo-Magarima districts. I have no helpful information about Imbonggu. Komo-Magarima could have been split from Nipa, Tari, or both.
16. Western Highlands: Hagen Central, Hagen North, and Waghi districts disappeared, and Anglimp-South Waghi, Dei, Mount Hagen, Mul-Baiyer, and North Waghi districts were created. Some districts must have been merged and others split, so any detailed description must be highly conjectural. Waghi district was presumably split into two parts, which became North Waghi and part of Anglimp-South Waghi. Hagen Central and Hagen North were redistributed in some way, so that part of Hagen Central became Mount Hagen. Another part of Hagen Central may have merged with Tambul to form Tambul-Nebilyer. Jimi district probably remained unchanged. Beyond that it would be rash to speculate.
17. Western: all of the districts - Balimo, Daru, Kiunga, Morehead, Nomad, and Tabubil - disappeared, and Middle Fly, North Fly, and South Fly were created. Based on geography, Kiunga and Tabubil must have merged to form North Fly; Balimo and Nomad merged to form Middle Fly; and Daru and Morehead merged to form South Fly.
18. West New Britain: probably Bialla and Kimbe districts merged to form Talasea; Gloucester and Kandrian districts merged to form Kandrian-Gloucester.

Other names of subdivisions:

1. Ambunti-Dreikikir: Ambunti-Drekikier, Ambunti-Drekikir (variant)
2. Anglimp-South Waghi: Angalimp-South Wahgi (variant)
3. Daulo: Asaro-Watabung (variant)
4. Finschhafen: Finschafen, Finschhafan (variant)
5. Huon: Huon-Gulf (variant)
6. Ialibu-Pangia: Kewabi Rural (variant)
7. Kandrian-Gloucester: Kandrian (variant)
8. Karimui-Nomane: Karamui-Nomane (variant)
9. Kiriwina-Goodenough: Kiriwini-Goodenough (variant)
10. Komo-Magarima: Komo-Margarima (variant)
11. Kompiam-Ambum: Kompiam (variant)
12. Koroba-Lake Kopiago: Koroba-Kopiago (variant)
13. Mendi-Munihu: Mendi (variant)
14. Mount Hagen: Hagen (variant)
15. North Waghi: North Wahgi (variant)
16. Sina Sina-Yonggomugl: Sinssina-Yonggamugi (variant)
17. Tari-Pori: Tari-Pari (variant)
18. Tewae-Siassi: Tawae-Siassi, Tewai-Siassi (variant)
19. Unggai-Benna: Unggai-Beni (variant)