George,
I believe the Barclay Center is suitable for hockey. However, it doesn't hold as many for hockey as other arenas, such as MSG.
I don't think they'll be changing their name.
They currently play and have played for years at the Nassau County Coliseum which is in Uniondale or East Meadow, not that far from Freeport. It's nicknamed the Mausoleum. It's old and I guess it's time has come. Attendance has been down. However, the possible max attendance at the Barclay Center (14,500) is not much higher than this year's attendance (approximately 13,000).
The owner Charles Wang has been trying to get a new arena with public support for years to no avail. The last effort came about two years ago when the voters of Nassau County turned down a tax that would have been imposed on all households. Before the vote, Wang announced that if the voters turned him down, he was going to move the team and he was true to his word. However, he wanted to keep it in the NY area and with the Barclay Center he can.