No one NEEDS to take pain medication. We take it because we cannot/will not mentally handle the pain. Not saying there is anything wrong with that but pain meds don't cure anything, they just make pain more bearable. And not everyone who is in pain should take opioids. We don't even know what the situation is with Whoseajiggawhaaa's mom is and what she is being prescribed the Oxycontin for (Is it for chronic (long-term) pain? Short-term pain? How much pain is she in and what is causing it? etc)
It is ignorant to say "addicts are the people you see when you drive downtown", as though the only addicts are visible down-and-out homeless people who are addicted to hard street drugs. I'm sure plenty of people on this forum who have addictions would strongly disagree with that assessment. There are tons of people who appear relatively normal who have addictions, and tons of people get addicted to Oxycontin.
OP, if your mother takes Oxycontin every day for more than a week or 2 she will get physically dependent on it. Sometimes people mistakenly use the term "addiction" to describe physical dependence, but they are not the same thing and you can have one without the other. Being physically or physiologically dependent means that the body has adjusted to the drug and now needs it to function normally, so when you stop taking the medication (or don't take it often enough or in a high enough dose) you will get withdrawal symptoms. Being addicted involves the person's behaviour, it means that you lack control over your use of the drug, you continue taking the drug even when you don't really want/need to or there are negative consequence, and you are preoccupied with the drug. Often people deny that they have a problem and hide their addiction from others. People can and do get addicted to Oxycontin, even when prescribed for pain.
It is impossible to know if your mom will get addicted, but it is not the drug itself that determines whether someone will become addicted, it is whether or not their brain is prone to addiction. If the person has depression, anxiety, or takes the drug in ways it wasn't prescribed, these are all signs that they might be more likely to get addicted.
What makes you say your mom does not have an "addictive personality"? Simply not having had a previous drug addiction does not prove anything, many people get addicted to opioids when they have no history of drug addiction. What is her mood like, her behaviours?
If your mom does not have any mental health issues (depression, anxiety, ADHD, etc), has never had any addictive behaviours around anything else (can be non-drug, like shopping, internet, gambling, etc), only takes her meds because she is in extreme pain, and uses them exactly as prescribed, then it is unlikely that she will develop a mental addiction to the Oxy.