I am not a native English speaker and while reading the Harry Potter books, I wasn't aware that "elder" is a plant. So I automatically assumed the elder wand was called "elder" on account of age (which generally implies power as well). Recently, I saw the movie (dubbed), and there it was translated to the term referring to the plant. So I learned that "elder" means both, and that got me thinking. Is there any canon (or otherwise) source on which meaning applies to the "elder wand?" Or is it ambiguous on purpose?
It's unambiguously referring to the wood.
So Death crossed to an elder tree on the banks of the river, fashioned a wand from a branch that hung there, and gave it to the oldest brother.
“Nah, that story’s just one of those things you tell kids to teach them lessons, isn’t it? ‘Don’t go looking for trouble, don’t pick fights, don’t go messing around with stuff that’s best left alone! Just keep your head down, mind your own business, and you’ll be okay’ Come to think of it,” Ron added, “maybe that story’s why elder wands are supposed to be unlucky.”
“What are you talking about?”
“One of those superstitions, isn’t it? ‘May-born witches will marry Muggles.’ ‘Jinx by twilight, undone by midnight.’ ‘Wand of elder, never prosper.’ You must’ve heard them. My mum’s full of them.”
To add to the other answers here, Pottermore's wand quiz makes Elder wood the hardest type to get, with a 00.5% chance of getting it.
Elder wood is an extremely rare type of wand wood.