In order to understand the observed physical and orbital diversity of extrasolar planetary systems, a full investigation of these objects and of their host stars is necessary. Within this field, one of the main purposes of the GAPS observing project with HARPS-N at TNG is to provide a more detailed characterization of already known systems. In this framework we monitored the star, hosting two giant planets, HD 108874, with HARPS-N for three years in order to refine the orbits, to improve the dynamical study and to search for additional low-mass planets in close orbits. We subtracted the radial velocity (RV) signal due to the known outer planets, finding a clear modulation of 40.2 d period. We analysed the correlation between RV residuals and the activity indicators and modelled the magnetic activity with a dedicated code. Our analysis suggests that the 40.2 d periodicity is a signature of the rotation period of the star. A refined orbital solution is provided, revealing that the system is close to a mean motion resonance of about 9:2, in a stable configuration over 1 Gyr. Stable orbits for low-mass planets are limited to regions very close to the star or far from it. Our data exclude super-Earths with Msini ≳ 5M⊕ within 0.4 AU and objects with Msini ≳ 2M⊕ with orbital periods of a few days. Finally we put constraints on the habitable zone of the system, assuming the presence of an exomoon orbiting the inner giant planet. Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF at the Spanish Observatorio del Roque de los Muchachos of the IAC in the frame of the programme Global Architecture of Planetary Systems (GAPS).Table A.1 is also available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/599/A90</A
