Going to Iceland at any time of year is a great idea. Each season has its own distinct charms and challenges.
December - Magical Christmas (Jól) season with the 13 Yule Lads starting December 12th
New Year's Day - Amazing fireworks display on top of the world
Northern Lights - Definitely visible in North Iceland and likely around Reykjavik
January - Winter sports at Bláfjoll ski area near Reykjavik; better skiing conditions up north near Akureyri
January to Mid-February - þorrablót - Mid-winter festival which features consumption of traditional Icelandic foods, such as svið (jellied sheep's head), hákarl (putrefied shark), and hrútspungur (ram's naughty bits). Yum! At least there is Brennivin (Icelandic schnapps) to wash it down.
Challenge - limited sunlight - 4 - 6 hours, hard to drive around country unless vehicle is properly outfitted.
Mid-April - First day of summer celebrated in Iceland
May - Driest month of the year statistically
Fewer tourists, less expensive to fly, drive and stay in Iceland
Holidays - Sjomannadagur (Seaman's Day, early June) - Entertainment and activities around Reykjavik's harbor; Independence Day (June 17) - Family-friendly day of free activities and concerts in downtown Reykjavik; Summer Solstice (June) - longest day of year features parties long into the night
Challenges - Many shops and businesses are closed around the Easter holiday; in North Iceland or West Fjords, some areas (ex. Dettifoss) and activities (ex. whitewater rafting) may not be accessible or available until the end of May).
Unlimited daylight, virtually 24 hours as sun never really sets.
Festivals, festivals and more festivals! Verslunnarmannahelgi (businessman's holiday, usually first Monday in August) - many people go camping around the country, with the Westman Islands a popular destination; Secret Solstice (June); Hinsegin Dagar (Gay Pride weekend, mid August); Menningarnótt (Culture Night, mid-August) - one of Reykjavik's largest festivals, with many free performances, street fairs, and fireworks; Reykjavik Marathon (mid-August); Rettir (Sheep and Horse Roundup, early September) - various locations
Challenge - By Icelandic standards, crowded, though if going away from major attractions in Reykjavik and the southwest corner of the country, should still be plenty of space. Of course, more expensive as well.
Less expensive, fewer crowds, and still relatively warm through mid-October
If lucky, especially so in the southwest part of the country, perhaps you can catch the Northern Lights. (I saw in Reykjavik in October 2005.)
Festivals - Iceland Airwaves (November) - Top notch independent music festival now since founding in 1999 at venues throughout Reykjavik; Reykjavik International Film Festival (September/October); Yule Lads (mid-December) - Starting December 12th and for 13 days thereafter leading up to Christmas, enjoy the Icelandic Yule Lads festivities; Reykjavik Jazz Festival (Jazzhátio Reykjavikur, September).
Challenge - Icelandic weather, always tricky, can get an early start on winter, so earlier in the season is better if you want to move around the country.