LISA, the first space-based gravitational wave observatory, will be able to survey the whole sky, providing for the first time insights on the physics of gravitational waves in the low-frequency band (0.1 mHz - 1 Hz). The three arms of the interferometer are defined by free-falling test masses (TMs). Each TM is a 46 mm Au-coated cube of Au/Pt. High-energy particles interacting with the spacecraft may induce a net charging rate in the TMs, which would result in acceleration noise. Monitoring the variations of the cosmic-ray flux and detecting the high-energy component of solar energetic particle (SEP) events will be essential to understand the charging background of the mission and to provide vetoes for fake gravitational-wave triggers. We present the design of a Radiation Monitor tailored to monitor the charging rate of the TMs. It aims at providing a high-sensitivity and low-power consumption solution for detecting protons and alpha particles at a few hundred MeVs. It will be able to observe SEP events and short-term variations of the cosmic-ray flux at 1 AU from the Sun, in an energy band that is inaccessible for most radiation monitors. It consists of a telescopic arrangement of absorbers and plastic scintillators coupled to arrays of silicon photomultipliers (SiPMs). The SiPM readout is performed with the BETA ASIC, which is capable of amplifying, shaping and digitizing up to 64 input signals with only $\sim 1$ mW/ch. We describe the initial design of the radiation monitor and discuss its expected performance, based on Monte Carlo simulations.