This is really a comment on Joel's answer, but apparently too long. Let P be the forcing which adds a permutation of $\mathbb{N}$ by finite pieces (so $P$ is forcing-equivalent to Cohen forcing). Force with a finite-support iteration of length $\omega_{1}$, using $P$ at each stage, over a ground model in which CH fails (producing, for example, the so-called dual Cohen model). The generic reals then ought to give a witness to the rearrangement number (the least cardinality of a set of permutations with the rearrangement property) being $\aleph_{1}$, while the continuum stays large. 

Reasoning by analogy, this suggests that the rearrangement number is bounded above by $\operatorname{non}(M)$, the least cardinality of a nonmeager set of reals (which seems to be the same as the least cardinality of a nonmeager set of permutations, but I haven't thought it through in detail). Indeed, a nonmeager set of permutations has the rearrangement property, since for any series which is not absolutely convergent, comeagerly many permutations witness this.