Oni (no relation)
"Oni" being a popular mythical concept in Japan, there are a number of works and characters titled (or abbreviated) "Oni". Due to the shared inspiration, interesting similarities with Bungie's Oni can sometimes be found.
Oni RPG series and OVA
A classic RPG series created by Pandora Box, and popular in Japan, depicted transforming heroes. It started on the Gameboy in 1990 (here's what one cartridge looked like), then moved to the SNES, the PlayStation (where an entry named "Oni Zero ~Resurrection~" released in 2001!; thankfully not for the PS2, which would have been confusing) and the Nintendo DS, with the most recent game having come out in 2007. A 1995 arcade fighting game called Oni: The Ninja Master (despite already being named in English using katakana) was released in the Western world with the inscrutable name Metamoqester.
Also in 1995, the game universe was loosely adapted as an OVA called "Touma Kishinden Oni" in 1995, consisting of 25 10-minute episodes (released on five discs). Incredibly obscure and hard to find, the opening is here. It's about a young warrior who has to come to terms with his inhuman nature and ability to transform into an "oni".
Halo's ONI
See Easter eggs#Halo.
AT-43's Oni
An RPG board game with a faction called Oni. It's an Umbrella-like corporation that managed to weaponize the zombie thing somewhat reliably; most of ONI's troops are cyber-zombies of some kind.
Akuma Oni in Street Fighter
Oni (sometimes called Akuma Oni or Oni Akuma for clarity) is a "demon" form of long-time Street Fighter character Akuma who has been corrupted by the dark energy of his martial art. He first appeared in Super Street Fighter IV. There is also a "Pandora's Box" version of Oni in Street Fighter X Tekken.
Also not Oni
Oxygen Not Included
No connection to the concept of the Japanese oni, but abbreviated to "ONI" often enough that it competes for attention with Oni in search engine results. We even had to change our Discord server's name to "Oni (Bungie, 2001)" to prevent confused Oxygen fans from joining.