Definition of a von Neumann algebra Is there a way to equip every C*-algebra A with a functorial topology such that
the canonical map A→A** is an isomorphism if and only if A is a von Neumann algebra?
Here A** denotes the dual of A* in some other functorial topology.
Of course if we restrict our attention to von Neumann algebras from the beginning,
then the σ-weak topology on A and the norm topology on A* answer the question.
 A: This isn't quite an answer, but it might lead to one.  Takesaki, III, Thm 3.16 shows that a C*-algebra A is a W*-algebra (i.e. non-spacial version of a von Neumann algebra) if and only if A is monotone closed and admits sufficiently many normal positive linear functionals.
Monotone closed == bounded increasing net of self-adjoints has a supremum.
Normal == Positive functional which respects the supremum.  I guess we can then define normal to mean: positive and negative parts etc. are normal.
So on the class of Monotone Closed C*-algebras, we could let A* be the space of normal functionals, and give A* to norm topology.  Then I think A=A** only when A is a von Neumann algebra (or otherwise A won't even inject into A**).
What I don't see is how to extend this to all C*-algebras.
A: Here's a topology that will work for separable von Neumann algebras:
Fix a separable Hilbert space H.
The topology on A is obtained by pulling back the σ-weak topology on B(H) along all possible C*-algebra homomorphisms A→ B(H).
Similarly, the topology on A* is obtained by pushing forward 
the norm topology on l1B(H) = (B(H)),σ-weak)*
along the maps l1B(H) → A*
induced by the above homomorphisms A→ B(H).

If you want this to work for all C*-algebras, then the answer to your question is probably ''no''.
Indeed, a non-normal representation of a von Neumann algerba A→ B(H) is a morphism of C*-algebras, and in particular should be continuous
for your topology (that's how I interpret your requirement "functorial topology").
This looks impossible to me.
A: Consider the monomorphism A→A** of Banach spaces.
Here A** denotes the second dual of A as a Banach space.
The Banach space A** is a von Neumann algebra with the predual being A*.
See Section 1.17 in Sakai's C*‑algebras and W*‑algebras.
We have a commutative square of Banach spaces consisting of morphisms
A→A**→B** and A→B→B**.
Thus we can pull back the ultraweak topology on A** to A
and obtain a functorial topology on C*‑algebras due to the commutativity of the square above.
Henceforth denote by A* the dual of A in the new topology and
by A** the dual of A* in the norm topology
If the canonical morphism A→A** is an isomorphism, then A has a predual,
therefore it is a von Neumann algebra.
Unfortunately, if A is a von Neumann algebra, then the functorial topology
does not coincide with the ultraweak topology
and the canonical morphism A→A** is not an isomorphism.
We can fix this problem by composing the monomorphism A→A** with the multiplication by a certain central projection.
However, the definition of this central projection relies on the fact that A is a von Neumann
algebra and I don't see any way to extend it to arbitrary C*‑algebras.
