It wasn't only that it was the day of Reagan's inauguration, it was minutes after he was inaugurated that it happened. Most people seem to agree that Khomeini waited until Reagan was president because Carter had refused to send the Shah back to Iran. The Shah was a pro western dictator that ruled Iran and was supported by the US. At the start of the revolution the Shah fled Iran and came to the US for medical treatment, while here Carter refused to send him back (for health concerns). This strained US and Iran relations and the students stormed the Embassy in retaliation.

Carter and his staff worked non-stop to bring the hostages home. Finally in September, Khomeini wanted to put an end to the crisis and many people thought Carter would pull off the "October Surprise" and bring the hostages home before the election (maybe even win the election with it) but negotiations were slow and even after Carter lost the election, they worked to get the hostages home. Finally securing their release in January. (http://www.pbs.org/wgbh/americanexperience/features/general-article/carter-hostage-crisis/)

Many people see the release of the hostages as symbolic because of Carters refusal to send the Shah back to Iran, they refused to release the hostages until a new president was in office. Many people feel Carter lost the election because of this, people saw him as weak and his work the release the hostages took away time from campaigning. Others believe that Reagan negotiated while still campaigning but we have found no evidence of this and Reagan always denied it. (http://www.history.com/topics/iran-hostage-crisis)