Why do most drugs (eg: oxycodone) have seemingly arbitrary names?
The reason why the generic name of most drugs have seemingly little semblance to their chemistry is in the interests of practicality; a doctor would find it easier to write sildenafil (generic name) or Viagra (brand name) in a prescription than 5-{2-Ethoxy-5-[(4-methylpiperazin-1-yl)sulfonyl]phenyl}-1-methyl-3-propyl-1$\ce{H}$,6$\ce{H}$,7$\ce{H}$-pyrazolo[4,3-d]pyrimidin-7-one (chemical name)  unless he had a weird fetish for IUPAC nomenclature .
As for how manufacturers (and the relevant regulatory bodies) arrive at those generic and brand names,  msdsmanuals.com has a take on this,

When a drug is under patent protection, the company markets it under its brand name. When the drug is off-patent (no longer protected by patent), the company may market its product under either the generic name or brand name. Other companies that file for approval to market the off-patent drug must use the same generic name but can create their own brand name. As a result, the same generic drug may be sold under either the generic name (for example, ibuprofen) or one of many brand names (such as Advil or Motrin).

Furthermore, they add,

Generic names are usually more complicated and harder to remember than brand names. Many generic names are a shorthand version of the drug’s chemical name, structure, or formula. In contrast, brand names are usually catchy, often related to the drug’s intended use, and relatively easy to remember, so that doctors will prescribe the drug and consumers will look for it by name. Brand names often suggest a characteristic of the drug. For example, Lopressor lowers blood pressure, Glucotrol controls high blood sugar (glucose) levels, and Skelaxin relaxes skeletal muscles. Sometimes, the brand name is simply a shortened version of the drug’s generic name—for example, Minocin for minocycline.

I found equally useful information on popsci.com,

As for commercial names, pharmaceutical companies are reluctant to share their strategies. These are hugely important brands, even after the patents expire--nobody knows what Sildenafil citrate is, but everyone knows its brand name of Viagra. And the odds are you know someone who insists on brand-name Tylenol instead of discount-store acetaminophen.
Though pharmaceutical companies contacted by PopSci wouldn't share their marketing secrets, you can still derive a drug's molecular and popular heritage just by studying its name. Like, for example, Tylenol. Its chemical name is $\ce{N}$-acetyl-p-aminophenol--see the -tyl and -ol in there? Then there's Viagra. It doesn't sound like sildenafil, but it certainly suggests vigor and virility--exactly what Pfizer wants you to think. Xeljanz, which was approved earlier this year after almost 20 years of research, targets a protein called Janus kinase. It works in a different cellular pathway from other arthritis drugs. That's where you get the -jan part of that name, which, to someone with a drug decoder, connotes its uniqueness.

As for oxycodone specifically, it's anyone's guess, since that information can't be found (online) easily enough.

 Note: I've italicized all generic and brand names because, well... they're generic and brand names. Italicizing actual chemical/IUPAC names is against convention, so don't get the wrong idea ;-)