I believe Reagan played a huge role in the release of the hostages in 1981. Reagan had just been sworn in by an America that was aching for a strong president. His reputation as someone who was likely to jump into action instead of waiting the situation out (some called him a "war monger") was what I believe finally brought the hostages to freedom.
I don't see how Reagan could have been too involved in the negotiations since he just took office the day they were released. Carter had been working on their release for two years before he left office. The timing was weird considering it was just minutes after Reagan took his oath. I believe that there were many people working on this before Reagan was in the picture. He just gets credit since it happened on his watch.
The sad thing is after Carter's administration spent all that time and effort negotiating for the hostage release, Reagan comes into office and takes full credit. It's another example of the smoke and mirrors politics of the GOP. They slant things to fit their agenda, while at the same time fooling and lying to the American people.
The hostage release was completely Jimmy Carter's project. Behind the scenes Carter did everything he could to try and get them released and had made headway through diplomatic channels. The only reason Reagan got the credit is because someone on the Iranian side of things decided to mock Carter and wait until the next president to capitulate. Reagan was a good actor and a horrible president.