Codeword stabilized (CWS) construction defines a quantum code by combining a classical binary code with some underlying graph state. In general, CWS codes are non-additive but become additive stabilizer codes if derived from a linear binary code. Generic CWS codes typically require complex error correction; however, we show that the CWS framework is an efficient tool for constructing good stabilizer codes with simple decoding. We start by proving the lower Gilbert-Varshamov bound on the parameters of an additive CWS code which can be obtained from a given graph. We also show that cyclic additive CWS codes belong to a previously overlooked family of single-generator cyclic stabilizer codes; these codes are derived from a circulant graph and a cyclic binary code. Finally, we present several families of simple stabilizer codes with relatively good parameters, including a family of the smallest toric-like cyclic CWS codes which have length, dimension, and distance as follows: $[[t2+(t+1)2,1,2t+1]]$, t=1,2, ...