We consider the problem of estimating the Hurst parameter for long-range dependent processes using wavelets. Wavelet techniques have been shown to effectively exploit the asymptotic linear relationship that forms the basis of constructing an estimator. However, it has been noticed that the commonly adopted standard wavelet estimator is vulnerable to various non-stationary phenomena that increasingly occur in practice, and thus leads to unreliable results. In this paper, we propose a new wavelet method for estimating the Hurst parameter that is robust to such non-stationarities as peaks, valleys, and trends. We point out that the new estimator arises as a simple alternative to the standard estimator and does not require an additional correction term that is subject to distributional assumptions. Additionally, we address the issue of selecting scales for the wavelet estimator, which is critical to properly exploiting the asymptotic relationship. We propose a new method based on standard regression diagnostic tools, which is easy to implement, and useful for providing informative goodness-of-fit measures. Several simulated examples are used for illustration and comparison. The proposed method is also applied to the estimation of the Hurst parameter of Internet traffic packet counts data.