Fascism grew out of the 19th C Romantic tradition which often honoured the ancients where people hadn't been corrupted by the virus of modernity. Its extreme nationalism had to be founded on a mythos of the intrinsic greatness of the nation, and in Italy the 'proof' was the Roman Empire. The name was intended to evoke this.
[Mussolini]called to the remembrance of his little group of ex-soldiers the symbol of the Roman power, the fasces, the staves of the lictors. He gave them the ancient Roman salute and made for them a scheme of military and political organization on the model of the old Roman legions; his Fascisti were divided into principi and triari, into maniples, centuries, cohorts, legions; and when they were marching on Rome, he made this solemn affirmation to them at Civitavecchia: "I swear to lead our country once more in the paths of our ancient greatness." The ideal has always remained the same, for he says, "We represent the spirit which once carried the legions of the consuls to the farthest limits of the earth," or again, "The example of Ancient Rome stands before the eyes of all of us, but the Colosseum, the Forum Romanum, only proclaim the glory of the past, and we have to found the glory of today and of tomorrow. (Mussolini and the Roman Empire - K Scott)