The consensus seems to be something like this answer from MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/LongDivisionSymbol.html.

The symbol separating the dividend from the divisor in a long division problem that is drawn as a right parenthesis (or sometimes a straight vertical bar) with an attached viniculum extending to the right seems to have no established name of its own. It can therefore simply be termed the "long division symbol," or sometimes the division bracket.

Your students may prefer the "gozinta" name suggested at WikiAnswers;-)