Most modern systems, be it cyber-physical or mere software systems, are highly configurable. The main challenge when dealing with such configurable systems stems from the usually huge number of system variants that can be exponential in the number of configuration options or features. Monitoring systems that react on observations, e.g., sensor data, varying across system configurations or being themselves configurable also face this challenge but have barely been considered in the literature. In this paper, we discuss new aspects for runtime monitoring with variability in the system being monitored as well as the monitor itself. As a first step towards a configurable-by-construction runtime monitoring approach, we introduce configurable monitors from an automata-theoretic and stream-based perspective. For this, we harvest existing work on featured transition systems and present a variability-aware variant of the stream-based specification language Lola.