In the current study, we explore stability of satellite motion around the natural moons of planets in solar system using the novel concept of ER3BP with variable eccentricity. This concept was introduced earlier when novel type of ER3BP (Sun–planet–satellite) was investigated with variable spin state of secondary planet correlated implicitly to the satellite motion (in the synodic co-rotating Cartesian coordinate system) for its trapped orbit near the secondary planet (which is involved in Kepler’s duet “Sun–planet”). However, it is of real interest to explore another kind of aforedescribed problem, ER3BP (planet–moon–satellite) with respect to investigation of satellite motion m around the natural moon m_moon of planet with variable eccentricity of the moon in its motion around the planet. Therefore, we consider here two primaries, M_planet and m_moon, the latter orbiting around their common barycenter on quasi-elliptic orbit with slow-changing, not constant eccentricity (on a large-time scale) due to tidal phenomena. Our aim is to investigate the motion of a small dot satellite around the natural moon of planet on quasi-stable elliptic orbit. Both novel theoretical and numerical findings (for various cases of trio “planet–moon–satellite”) are presented in the current research.