We present the analysis of multiband time series data for a sample of 24 Cepheids in the field of the Large Magellanic Cloud cluster NGC 1866. Very accurate BVI Very Large Telescope photometry is combined with archival UBVI data, covering a large temporal window, to obtain precise mean magnitudes and periods with typical errors of 1-2 per cent and of 1 ppm, respectively. These results represent the first accurate and homogeneous data set for a substantial sample of Cepheid variables belonging to a cluster and hence sharing common distance, age and original chemical composition. Comparisons of the resulting multiband period-luminosity and Wesenheit relations to both empirical and theoretical results for the Large Magellanic Cloud are presented and discussed to derive the distance of the cluster and to constrain the mass luminosity relation of the Cepheids. The adopted theoretical scenario is also tested by comparison with independent calibrations of the Cepheid Wesenheit zero-point based on trigonometric parallaxes and Baade Wesselink techniques. Our analysis suggests that a mild overshooting and/or a moderate mass-loss can affect intermediate-mass stellar evolution in this cluster and gives a distance modulus of 18.50 +/- 0.01 mag. The obtained V,I colour-magnitude diagram is also analysed and compared with both synthetic models and theoretical isochrones for a range of ages and metallicities and for different efficiencies of core overshooting. As a result, we find that the age of NGC 1866 is about 140 Myr, assuming Z = 0.008 and the mild efficiency of overshooting suggested by the comparison with the pulsation models.