In this work, we carry out a comprehensive analysis for the weak-value-amplification (WVA) measurement in a single trapped Ca+40 ion system, for arbitrary measurement strengths. In addition to carry out a compact expression for the meter's shift, we pay particular attention to the changes of the distribution profile of the output data and its variance width. With them, we are able to characterize the quality of the WVA measurement by means of the signal-to-noise ratio criterion and the Fisher information. For both types of characterizations, we compare the quality of the WVA scheme with that of the conventional measurement in detail. Viewing the versatility of phase tuning of the driving field, we also investigate the imaginary WVA measurement. All the extended analysis and results in this work can be examined by the state-of-the-art experiments in the trapped ion systems, which may shed light on possible applications in quantum technologies.