At the very least they would need to find something to legitimise their claim. Recapturing some of the artefacts from the WRE - or offering to retake Western Africa for the Pope in Rome, both of these would be big bargaining chips to get the Pope to grant the title. For whatever purpose it may have.
Any way that could suggest legitimacy is the lynch pin. Perhaps descent from one of the Eastern Roman Emperors? They had branch families that survived in Italy, it would be very tenuous, but then at least there can be a claim that they descend from the Imperial Purple.
The only issue is that the Pope won't be willing to let the Neo-WRE exist without being strong-armed. Technically the Pope would have to acknowledge his superiority - and therefore would want to award the title to a weaker HRE, rather than the strong Neo-WRE. A campaign to take and unite Italy however - marching into Rome may be unpopular with the other Kings - but that would certainly assure the Pope agrees. Although, there could always be a schism.
Man, I don't think I even know half the complications that resurrecting that title would cause to Europe. Especially as it gives a loose legitimacy to reconquer the Med. Perversely, the Pope might encourage that to avert being strong-armed. Take Egypt, Take Greece, etc - this could lead to colossal wars if the Pope matters to the Neo-WRE.
I look forward to the new Capital though, you can't well be the Western Roman Empire in Paris or Madrid - Rome itself is a great nominal capital, but a real Administrative Capital in southern France could be a beautiful place. Neo-Roman frippery, combined with modern-ish technology. I can just see a cannon decorated with an Eagles head, or a royal guard in legionary kit.