This question can be approached abstractly through the general Tannakian formalism,
as laid out e.g. here, or very concretely by hand. You construct maps in both directions. To a $G_m$ gerbe assign its tensor category $QC(Gerbe)$ of sheaves (I'll speak of quasicoherent sheaves out of force of habit - presumably you can work just with coherent). This is a commutative algebra over $QC(X)$, which locally is isomorphic to graded sheaves on $X$ (i.e. to $QC(X)\otimes Rep G_m$). Conversely to such a category assign its spectrum, the stack which to any ring attaches the groupoid of tensor functors from your category to modules. This carries a map to $X$ which is a $G_m$ gerbe.
Put another way, given a sheaf of tensor categories over $X$ (or commutative algebra over $QC(X)$) which is locally isomorphic to $Rep G$, you consider the stack (sheaf of groupoids) of isomorphisms of this sheaf of categories with $QC(X)\otimes Rep G$ [EDIT: Better and more Tannakian to say, the stack of fiber functors to $QC(X)$]. This is a $G$ gerbe (ie 's locally of the form $X\times BG$). This is of course just the usual Tannakian reconstruction as in Deligne, except that we have the base $X$ be a scheme (or geometric stack) instead of the spectrum of a field.
Of course you could also ask to give a more global characterization of such $Z$-graded commutative algebras over $QC(X)$. I think it's equivalent to characterize the module category given by sheaves of degree 1, aka twisted sheaves on the corresponding gerbe. This is your version of a categorified line bundle -- it's a module category locally isomorphic to sheaves. Presumably it can be characterized as an invertible module category -- one for which there exists (or maybe for which you specify - I'm conveniently pretending that everything has been taking place one level of categoricity down, which is fine if you only care about say a class in $H^2(X,G_m)$) an inverse with respect to tensor product of module categories over $QC(X)$. Then the above argument proves that such categorified line bundles (via Spec of the $Z$-graded $QC(X)$-algebra they generate) are equivalent to $G_m$-gerbes. [Edit:] You'll also need to make sure such invertible modules are locally trivial, ie again given by the same cohomology group. You might also want to think of things through a third perspective on this story after the gerby/Tannakian ones, namely that of Azumaya algebras. You want to know that your invertible module category has a generator as an $O_X$-linear category, and thus is equivalent to the category oif modules over a sheaf of algebras, namely the endomorphisms of this generator. And then appeal to a classification of these algebras up to Morita equivalence by the same cohomology group.
[By the way a very interesting recent paper about the derived version of this story is here.]