The wonderful thing about this subject is that Noriko T. Reider and Utah State University Press published her entire detailed, scholarly book on oni as a free pdf under a Creative Commons License. Japanese Demon Lore: Oni from Ancient Times to the Present is ready for your reading or perusing. Reider’s introduction includes a long discussion of the traditional characteristics of the oni. (pp. 7-9)

Oni are customarily portrayed with one or more horns protruding from their scalps. THey sometimes have a third eye in the centre of the forehead, and varying skin color, most commonly black, red, blue, or yellow. They often have large mouths with conspicuous canine teeth. More often than not, oni are scantly clad, wearing a loincloth of fresh tiger skin. The combination of the horn and the tiger skin may trace its origins back to an ancient folk belief. The image of oni with an ox horn(s) and tiger skin loincloth is said to have come about from a play on the word ushitora. Ushi (ox) represents the direction thirty degrees east from due north (north-north-east); tora (tiger) is the direction thirty degrees northward from due east. Ushitora was considered an ominous direction called kimon oni’s gate. Hence, ox horns and tiger skins are used to depict oni (Baba 46–47; Toriyama 80).

The Toriyama in that reference is Toriyama Sekien (1712-1788), author and artist of The Illustrated Night Parade of a Hundred Demons, an encyclopedia of strange and supernatural creatures. Toriyama believed that this play on words was the origin of the oni’s shape. “It is said that the shape was made by combining an ox and a tiger”. (quoted, p. 98, Reider).

This play on words may explain the combination of horns and tiger skin, but as we'll see, the horns by themselves go back farther.

Continuing with the oni morphology, Reider writes:

Oni are often depicted carrying an iron rod used to torture their human victims.

Although descriptions of oni vary, there are certain common physical traits that tend to be relatively constant from one representation to another, such as their three-fingered hands. As they are described in Konjaku monogatarishū (Tales of Times Now Past, ca. 1120), in the medieval period, the oni usually have only two or three fingers and toes with long and sharp nails. Indeed, some oni depicted in Jigoku sōshi (picture scrolls of Buddhist Hell, ca. 12th century), * Gaki sōshi *(picture scrolls of hungry ghosts, ca. 12th century),14 and *Kibi daijin nittō emaki *(Minister Kibi’s Adventures in China, late 12th century) are depicted with three toes (see Figure 1).

Figure 1 is a panel from that scroll, which you can view online in great detail courtesy of the Museum of Fine Arts Boston. (It’s the panel with the red monster walking towards a buiding. He has one horn in this example, though not a striped loincloth.)

Oni are frequently considered to have imposing physical stature. This may have some relationship to Orikuchi’s explanation of oni’s etymology, which is that the term was used to describe gigantic people. According to Fusō ryakki, a 12th century history book, on the twenty-fifth of the Fourth month of 929, a man saw an oni who was taller than the beam of a house (see Kōen 688). According to a story included in Konjaku monogatarishū, an oni that a man encountered at Agi Bridge in Ōmi province was nine feet tall.

Sometimes oni are described as hairy. According to (Yashiro-bon) Heike monogatari (Yashiro version of the Tale of the Heike, ca. early 13th century), the oni who attempts to kidnap Watanabe no Tsuna (953–1025) to Mt. Atago has a black arm full of white hair, and an oni described in Ōkagami (The Great Mirror, ca. 1085–1125) has “a shaggy hand with long, knifelike nails.” As mentioned earlier, a primordial form of oni is female, and there are a number of notably fierce female oni in Japanese literature and performing arts. But the popular image of oni remains predominantly male.

The image of the oni s likely not Japanese in origin. "Oni" itself is a Japanese word and concept, but in ancient Japanese texts, the oni is originally is described as an invisible and malign influence. The native Japanese oni seems to have taken flesh from Chinese and/or Buddhist models.

Reider explains:

Kosugi Kazuo asserts that the root of the visual image of Japanese oni is found in Chinese gui-shen) (ghosts and spirits), and the oni’s shape has remained surprisingly unchanged from that of its Chinese predecessor. The gui-shen was originally an indigenous Chinese being and had no foundation in Buddhism. Buddhism, however, seems to have assimilated gui-shen into its pantheon. According to Kosugi, Chinese gui-shen came to Japan with Buddhism, and gave a shape to what was heretofore a shapeless oni. Kosugi finds evidence that indigenous Japanese concepts of the oni bore no clear visual image by merit of fact that when the Chinese gui-shen were introduced to Japan around the seventh century the Japanese applied the form to oni with little apparent change (Chūgoku bijutsushi 188–206).

• p. 9

For a related theory of origin, Reider mentions the argument of Orikuchi Shinobu who believed Japanese oni were originally aspects of kami, the gods.

After Buddhism was introduced in Japan, he continues, oni became mixed up with such Buddhist creatures as rasetsu (raksasa), gozu (ox-headed demons), and mezu (horse-headed demons).

The horns on some of those creatures may thus have influenced the depiction of the oni.

Do the horned oni have any connection with the horned demons of the Western tradition? I can see from googling that many people are tempted to search for some common ancient archetype of horned powerful supernatural figures. There are indeed horned gods and other supernatural creatures across the world. I’m not a medievalist, so I woudn’t even begin to speculate on the exact evolution of the Western iconography of the horned devil. (I do know those medievalists are the people to ask about it, and that scriptural imagery of goats is involved.) But even if there is some common cultural connection of horns and powerful frightening entities, Japanese oni and the Western devil would be like 30th cousins several times removed. They evolved far from each other in different contexts.

Personally speaking, perhaps a good reason for the terrifying oni to have those horns is simply that ox’s horns are dangerous and scary. I never appreciated how scary until one day hiking in the Bavarian Alps. One of those beautiful red Alpine cattle herds with the cowbells round their necks emerged from the mist. It was a lovely sight until one of their number rushed towards me and prodded me with its horn. No harm done, but I was traumatized forever. Cattle horns can gore, and the oni are well set up to gore with their horns, as well as shred with their claws, and bite down on their human meals.