The standard example here is where the braided tensor category is the Drinfeld center Z(C) and the algebra object is the induction of the trivial object from C to Z(C). If C is semsimple over an algebraically closed field then this can be written explicitly as $\sum_x x \otimes x^*$ with half braiding given by Theorem 2.3 of Kirillov-Balsam.

There are plenty of trivial examples when the category is allowed to be symmetric (which presumably you don't want), for example any ordinary commutative algebra is an algebra object in the symmetric (and hence braided) tensor category Vec.