Before Christianity was ever centered in Rome, it spread throughout the Middle East, made its way across the former Persian empire to India and traveled south, deep into Africa. Although Rome was the center of the Roman empire, Roman culture was not shared by large swathes of the ancient Christian world. However, the Christian practice of venerating saints existed even among the ancient Christian communities in the Middle East, Far East and Africa (many of these communities still exist today). It is highly unlikely that a specifically Roman pagan practice would have been adopted across the entire ancient Christian world.

Info Source: "The Story of Christianity" by David Bentley Hart

So questions about the cult of the saints have come up a lot recently, and this one may be of particular interest to you. As far as the relationship between the cult of the saints and the cult of heroes, I'll defer to Peter Brown, who speaks directly to the question in the first chapter of his book The Cult of the Saints: Its Rise and Function in Latin Christianity:

Pagan parallels and antecedents can only take us so far in understanding the Christian cult of saints, very largely because the pagan found himself in a world where his familiar map of the relations between the human and the divine, the dead, and the living, had been subtly redrawn.

Let us take one well-known example: the relation between the ancient cult of the heroes and the Christian cult of the martyrs. To idealize the dead seemed natural enough to men in Hellenistic and Roman times. Even to offer some form of worship to the deceased, whether as a family or as part of a public cult in the case of exceptional dead persons, such as heroes or emperors, was common, if kept within strictly define limits. Thus, the practice of "heroization," especially of private cult offered by the family to the deceased a a "hero" in a specially constructed grave house, has been invoked to explain some of the architectural and artistic problems of the early Christian memoria. But after that, even the analogy of the cult of the hero breaks down. For the position of the hero had be delimited by a very ancient map of the boundaries between those things who had been touched by the taint of human death and those who had not: the forms of cult for heroes and for the immortal gods tended to be kept apart. Above all, what appears to be totally absent from pagan belief about the role of the heroes is the insistence of all Christian writers that the martyrs, precisely because they had died as human beings, enjoyed close intimacy with God. Their intimacy with God was sine qua non of their ability to intercede for and, so, to protect their fellow mortals. The martyr was the "friend of God." He was an intercessor in a way which he hero could never have been.

Peter Brown's argument is that the cult of the saints was not simply the product of popular religion, a simple changing of names of gods and heroes to saints. Instead, the cult of the saints is the product of changing beliefs about the relationships between the living and the dead. While there were certainly parallels with older traditions, and funerary customs and similar pre-Christian rites did continue at grave sites for much of the fourth and fifth centuries (which caused church leaders no small amount of anxiety), the role saints play in the cosmological relationship between the living, dead, and God was markedly different to the extent that I do not believe it would be useful to say the one naturally sprang from the other. The borrowing of certain cultural elements is certainly inevitable, and so I do not think we can (or should) completely throw out the role of pre-Christian funerary practices, but the practice of venerating saints and the holy martyrs has a distinct flavor unique to the developing theology of the dead.