The first question is in a system supporting anyons, what are the universal/topologically invariant physical quantities? Our current theoretical understanding is that all such quantities are related in one way or another to anyon braiding, to be more precise, the $S$ and $T$ matrices. Here $T$ matrix is a diagonal matrix whose diagonal entries are the topological twist of anyons (roughly speaking the exchange statistics), and the element of $S$ matrix represents mutual braiding of anyons. All the other invariant quantities, such as fusion rules and quantum dimensions can be derived from $S$ and $T$. For example, one can obtain fusion rules by applying Verlinde's formula from the $S$ matrix.
This does not mean that braiding is the only thing one can do for quantum computing using anyons. One can also fuse anyons and measure the outcome, or do inteferometry measurement. For example, just using measurement one can perform "braiding" without actually moving anyons physically, as explained in http://arxiv.org/abs/0808.1933. In some (potentially physically relevant) anyon models, braiding alone is not sufficient for universal quantum computation, and by adding measurement one can make it universal, for example http://arxiv.org/abs/1504.02098 and http://arxiv.org/abs/1405.7778 for the example of $SU(2)_4$ anyons.
Another interesting idea is to seriously make use of the topology of the manifold. $S$ and $T$ matrices are in fact related to modular transformations of the manifold. Roughly speaking, it is almost impossible to directly access the $T$ matrix of anyons with just braiding, while the so-called "Dehn twist" of the manifold can do so. This forms the basis of a proposal http://arxiv.org/abs/cond-mat/0512066 that attempted to make the Ising anyon model computationally universal. Of course, dynamically changing the topology is very difficult in reality, but still a quite beautiful idea. A recent refinement is to introduce geometric defects into the system, which serves as "branch cut" for anyons and effectively changes the topology. Braiding such defects therefore realizes Dehn twist, see http://arxiv.org/pdf/1208.4834.pdf. A broader context is that one can have various symmetry defects in an anyon system, which have distinct and possiblly richer braiding statistics. For example, defects in an Abelian anyon system can behave like non-Abelian anyons. For a general discussion see http://arxiv.org/abs/1410.4540, and for Abelian phases http://arxiv.org/abs/1305.7203.