Since this has been up long enough, I'll take a stab at it. The first place you're going to want to go is the Historia Langobardorum, which is where a lot of our information comes from. As far as I am aware, there is no evidence they claimed any sort of imperial derived power, and indeed, any semblance of 'Imperial' de jure authority over Italy and connection to Roman power structures ended with the fall of the Ostrogothic Kingdom. It is debatable whether Theodoric was interested in 'reviving' the Western Empire, or simply looking for ways to limit the theoretical authority of Eastern emperors over him. In the case of the Lombards, what I see is mostly opportunism, not long term planing for a great restoration. Indeed, the Lombards couldn't even completely subdue the Peninsula, let alone menace the other Germanic successor states in the West.
If anything, I'd say the Lombards aspired to the role as successors to Theodoric's Kingdom of Italy, an aspiration they couldn't come close to reaching, let alone successors of the Roman Empire.
Not in the slightest.
First you have to keep in mind that to the Lombards (as well as much of western europe) before Charlemagne, there was no "Western" Roman Empire. There was just the Roman Empire with its Roman emperor in Constantinople, and they were still around. In fact not just still around, they were right next door in Ravenna, Rome and Sicily. This is why the province around Ravenna even today is called Romagna. It was the land of the "Romans."
Because of this proximity, Lombard political identity formed in direct opposition to this "Roman" identity, so much that the bulk of the population under Lombard rule who were clearly descended from the Romans of antiquity, subsumed themselves into the Lombard identity and practically disappeared from the legal record by the 8th century.
This despite the fact that from a structural standpoint, there is quite possibly a bit of continuation from the late Roman past in the Lombard reliance upon law and the city for their administration.
I always like to tell people when explaining the problem of the "continuation of Roman identity" is you have to explain what Roman means at a particular place in time. What Roman means to us (the Colosseum, the centurion helmet, the toga) is not what Roman meant to the Byzantines (Orthodox Christian) is not what Roman meant to the Lombards and other successor nations (Greek).
Sources:
Christie, Neil. The Lombards: The Ancient Longobards. Oxford, UK: Blackwell, 1995. Print.
Wickham, Chris. Early Medieval Italy: Central Power and Local Society, 400-1000. Totowa, NJ: Barnes & Noble, 1981.