Hey there Ingolfur,
There really is no best time to come to Iceland; every time of the year has its own appeal and charm. If you come in midwinter, the weather will be cold and volatile, but you will have multiple opportunities to see the Northern Lights, as there are just a few hours of sunlight a day, and to explore the ice caves under Vatnajökull glacier, one of Iceland’s most popular activities.
If you come in midsummer, you lose out on these opportunities, but gain a lot more. You can camp in summer, take multi-day hikes throughout the highlands, reliably see puffins and great whales, and engage in many tours that do not run or are unreliable in winter. The weather (while still unpredictable) will also be far more pleasant.
Coming in the ‘low’ periods between the summer and winter often allows you to engage in activities from both seasons. In September, for example, you can very often still access the Highlands, still go puffin watching, but also take a Northern Lights tour, as it is getting dark enough to see them once more.
If you come outside of the summer months and Christmas/New Year period, flights are also likely to be much cheaper.
I recommend you look at the articles ‘Iceland in insert season here’, which provide comprehensive guides as to what to see, do and know about Iceland in each month. The link to Iceland in January is below, and you can follow the links to see to the rest.
Hope that helps,