It's not just a matter of it being old fashioned, like being named Octavia (going with the ancient fashion of naming your kid by their birthing order); or an old fashioned name people don't really like anymore such as Clementine, Ursula, or Prudence; or even as some people said "country bumpkin-ish names such as Billy Jo.
This name was picked because her character's grandfather was old fashioned, yes, but he was mainly angry for not getting the grandson he wanted, so he picked a name with a bad meaning out of spite.
In Korean her name could mean lots of things. 삼 means three/third and 순 can mean a few different things such as: sprout, purity, gentleness, or martyrdom. And yes, this is an old fashioned style of naming your kid...who wants to be named Third Sprout or Third Purity or Third Gentleness...if you take the nicest of translations. lol
But later on when you see her fill out the form to change her name, we see the chinese spelling is 三殉 which means the actual translation of her name is "Third Martyrdom".
It's like naming your kid Pain, Unhappiness, Suffering, or Sadness. Imagine being named that. Once you delve into what the name really means, who can blame anyone for not wanting such an awful name? I sure don't.
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