I'm 99% sure that changing names will have no effect whatsoever on how popular the club is.
The name can have a massive effect, and the 4 RU regions are living proof of that.
As I mentioned, the Dragons have people who won't go because they've got Newport in the name, and others who won't go because they've got Gwent in the name. Furthermore, there are people who won't go if one or the other name is lost from the team.
Valleys folk won't come down to Cardiff to watch the Blues and cite the fact that they're called Cardiff as the reason (there are others, but that's another discussion), whilst the city folk won't go if Cardiff isn't in the name, because they're the capital city.
The Ospreys got rid of the 'Neath-Swansea' part of their name as quickly as they could, as they realised there was no way they could build a fanbase by relying on supporters of two long-time rivals, and essentially created a new brand and a new fanbase.
Even the Scarlets, who were always effectively a regional team anyway, have plenty of people who don't like the fact that they dropped 'Llanelli' from their name, despite the fact that they were a new entity that sat above the old Llanelli club.
Personally, my feeling is that they need to act like representatives of Neath, or at the widest Neath Port Talbot. Give them a smaller catchment area that they can actually hit in depth and the local area are more likely to get behind them.