My research is in combinatorics and graph theory, and I'm used to phrasing basic combinatorics questions in terms of balls and bins. However, many sources talk about balls and urns. I'd never thought about it until a student brought it up this semester. They asked me why mathematicians are dumping the ashes of their loved ones in order to store coloured balls! I don't have an answer, and I agree that outside of a math class I've only heard urn refer to a vessel for cremated remains.
The linked Wikipedia article mentions Jacob Bernoulli's use of the Latin word urna, and its possible roots in old lotteries and elections. Are there are particularly good reasons why the word has persisted in English mathematics? Or is it just another curious survivor like singular?
The ancient Greeks used urns instead of ballot boxes. See, for instance, this for a learned discussion and nice picture.
There's also coffee urns, without which we could not prove theorems.
Added 15 Dec:
Humor aside, there are 2 or 3 things going on in the original poster's question, none really mathematical.
First, culture in general, and mathematical terminology in particular, changes in unpredictable ways, exhibiting both surprising conservative features and their opposite. One conservative mechanism is the role of culture heros and hero literature: if Bernoulli or William Feller used "urn" in the sense in question, so will we, to some extent.
Second, to be a math student is like visiting foreign country, where many things seem different. Many math words are wierd: why "field" in algebra, or "Gaussian kernel" in probability (what is nut-like about the latter?); to ask why we use these words is like asking why they drive on the wrong side of the road in some countries, or why parking spots are paid for in a different way, or why food names don't match up to one's expectations. The answer: just like foreigners, mathematicians are different, and only sometimes can one figure out how or why, so don't sweat it.
Another thing going on is ignorance of their own language: if your students read the dictionary more, they would realize that "urn" is not exclusively a funerary urn. Vanity insists, look at this image. The big cylinders with coffee in hotels and bad restaurants are usually called coffee urns: try googling the phrase. A pot or carafe is smaller, small enough to put on a diner's table.