According to Periodic Tales by Hugh Aldersey-Williams, every element has a use, or at least most of them do. The rare earth element, europium is used in European banknotes for security, along with TVs and fluorescent light bulbs. It is responsible for the unique red and blue fluorescence seen when the 5 euro note is illuminated with UV light.
Europium is one of the rarest elements in the universe. It makes up only about 5×10−8 % of all matter in the universe.
Is the use europium in the euro fortuitous, or does it have a purpose? It is probably a little of both. The chemist who added europium to the euro clearly enjoyed the accomplishment, but it also adds a unique security feature to Europe’s banknote.
By the way, if you live in Europe, don’t study the banknotes carefully, since it is apparently against the law to investigate the fluorescence of banknotes. See Suyver, Europium Secures the Euro, 2002 for more information on the fluorescence of the euro.
December 2, 2016: The real reason that Europe decided to use europium may be that they could not find sufficient supplies of unicornium.