I would assume that the Brazilian real is named after the Portuguese real which does mean "royal". It is an odd name in a democratic country, though.
I think that probably more accurately describes their intention behind the meaning of the name.
When I lived in Brazil in the 1990's, shortly after the currency changed from Cruzeiros to reais, the exchange rate was around 1.5. Today, it's 1.69 - there have been some relatively mild fluctuations since then but it has stayed remarkably stable.
http://en.wikipedia.org/wiki/History_of_the_Empire_of_Brazil...