  For Brylinski-Deligne covering groups of an arbitrary split reductive group,
we consider theta representations attached to certain exceptional genuine
characters. The goal of the paper is to determine when a theta representation
has exactly a one-dimensional space of Whittaker functionals, in which case it
is called distinguished. For this purpose, we first give effective lower and
upper bounds for the dimension of Whittaker functionals for general theta
representations. As a consequence, the dimension in many cases can be reduced
to simple combinatorial computations, e.g., the Kazhdan-Patterson covering
groups, or covering groups whose complex dual group is of adjoint type. In the
second part of the paper, we consider covering groups of certain
simply-connected groups and give necessary and sufficient condition for the
theta representation to be distinguished. There are subtleties arising from the
relation between the rank and the degree of the covering group. However, in
each case we will determine the exceptional character such that its associated
theta representation is distinguished.
