EEG power measurements have shown correlations with human language abilities, though the results have varied. Alpha power, believed to be modulated by thalamo-cortical networks, reflects sensory information processing. Additionally, alpha power has been more distinctly associated with language abilities compared to other EEG frequencies. This could be due to different degrees of reliability across frequency bands. The aim of this study was to identify the most reliable power measures and investigate their relation to language abilities. A group of children (N = 37) and adults (N = 90) participated in a study on app-assisted learning of a second language vocabulary. Participants completed a test on vocabulary and a resting-state EEG recording (180 s eyes open) before and after learning a second language using a smartphone app. Test-retest reliability of the EEG measures was quantified using Intraclass Correlation Coefficients (ICC). The measures with satisfying reliability (ICC > 0.75) were correlated with the vocabulary test. Highest ICC values were found in the alpha and beta range. Calculations yielded ICC values of 0.89/0.81 (children/adults) for occipital alpha power and 0.89/0.81 for beta power. These power values were correlated with the vocabulary test. Only within the adult sample was the occipital alpha power positively correlated with vocabulary (r = 0.24, p < 0.05). Alpha power could represent a stable neural marker for second language acquisition in the adult brain. The neurobiological components reflected by alpha power might exhibit considerable stability once fully developed, thereby leading to a reliable association with language across various studies.