Context. With this work we start a systematic analysis of the distant activity of several long-period comets in order to investigate the evolution of activity throughout the solar system and explore differences between comets that pass their perihelion at far or very close distances from the Sun. Aims: We present observational data for eight long-period comets, observed for the first time beyond r = 5 AU. Three targets have been characterised on their inward orbital branch. The others have passed their perihelion at quite large heliocentric distances (rq from 4.5 to 7.5 AU). Methods: We analyse multicolour broadband images (V,R, and I filters) taken at the Telescopio Nazionale Galileo to characterise the dust coma of the comets and investigate their morphology, photometry, colours, and dust production. Results: The morphological analysis shows many differences among the sample, from the large twisted structure present in the coma of comet C/2005 L3 to the regular coma envelope of C/2010 R1. The colour of the dust coma of all the comets is redder than the Sun. The Afρ value (measured in a reference aperture of radius ρ = 104 km) ranges from 114 ± 2 (C/2005 S4) to 5091 ± 47 (C/2005 L3) cm, depicting a scenario of bodies from moderately to very active. This is confirmed by the first-order quantitative estimate of the dust mass-loss rate for the comets that was obtained from the photometric data: assuming a grain velocity of v = 20 m/s, the dust production rate is comparable with, or even significantly larger than, that measured for many short-period ("old") comets at much smaller heliocentric distances. Based on observations collected at the Italian Telescopio Nazionale Galileo (TNG), operated on the island of La Palma by the Centro Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.