Via the boundedness of intrinsic g-functions from the Hardy spaces with variable exponent, {\mathcal{H}}^{p(\cdot )}({ℝ}^n), into Lebesgue spaces with variable exponent, L^{p(\cdot )}({ℝ}^n), and establishing some estimates on a discrete Littlewood-Paley g-function and a Peetre-type maximal function, we obtain several equivalent characterizations of {\mathcal{H}}^{p(\cdot )}({ℝ}^n) in terms of wavelets, which extend the wavelet characterizations of the classical Hardy spaces. The main ingredients are that, we overcome the difficulties of the quasi-norms of {\mathcal{H}}^{p(\cdot )}({ℝ}^n) by elaborately using an observation and the Fefferman-Stein vector-valued maximal inequality on L^{p(\cdot )}({ℝ}^n), and also overcome the difficulty of the failure of q = 2 in the atomic decomposition of {\mathcal{H}}^{p(\cdot )}({ℝ}^n) by a known idea.