It depends how the remaining axioms of ZF are altered (or not). For example, do we broaden the scheme of replacement to apply to formulas of set theory which contain the symbol "$arb$"? If so, then as Zhen Lin says the resulting theory proves the axiom of choice: given a collection of sets $\mathcal{A}=\lbrace A_i: i\in I\rbrace$, applying replacement to the formula "$x=arb(S)$" as $S$ ranges over the elements of $\mathcal{A}$ yields a choice function. If we don't do this - that is, if we have replacement and separation only for sentences of the language of set theory without "$arb$" - then we do in fact get a conservative extension: given any model $\mathcal{M}$ of ZF, let $arb$ pick any element from each element of $\mathcal{M}$ (I guess I'm using Choice in the ambient universe to do this, but if all we care about is not proving any new theorems, then this can be gotten rid of by Shoenfield absoluteness); the resulting extension is a model of ZF with $arb$.