Jan Mayen, where it is

On the island of Jan Mayen, belonging to Norway and located in the ocean between Greenland, Iceland and Norway, the climate is sub-polar, wet and windy, with a long, freezing and snowy winter, and a cold summer.
The average temperature is below freezing (0 °C or 32 °F) from November to April, and around freezing in May and October, while it rises a few degrees above in summer, between June and September.
Here are the average temperatures.
Jan Mayen - Average temperatures
Min (°C)-8-9-9-6-20331-2-5-8
Max (°C)-3-3-3-1146752-1-3
Min (°F)181616212832373734282318
Max (°F)272727303439434541363027

Snowfalls are frequent from October to May. The wind blows hard, especially from September to May, with real storms. June and July are the least windy months.
Beerenberg volcano, 2,277 meters (7,470 feet) high, is covered with perennial snows already from an altitude of around a few hundred meters.
Precipitation amounts to about 700 millimeters (27.5 inches) per year and is frequent and well distributed over the seasons. However, there is a relative maximun in autumn, and a relative minimum between April and July.
Here is the average precipitation.
Jan Mayen - Average precipitation

Being that it's above the Polar circle, in Jan Mayen you can see the midnight sun from May 13 to July 31, and the white nights (when the sun falls below the horizon, but it doesn't get completely dark) from April 8 onwards, and until September 4.
In winter, the sun never rises from November 20 to January 20, although around noon, when the sun is near the horizon, there's a bit of light even at the winter solstice (i.e. December 21, the shortest day of the year).

Winter goes from November to April and is characterized by low temperatures, a few degrees below freezing, frequent wind and snowfalls occurring almost daily, though usually not abundant. The average temperature in January and February, -6 °C (21 °F), is not so far from that of the Norwegian capital, Oslo. During some periods, the temperature can rise slightly above freezing, up to 1/2 °C (34/36 °F) even in winter, and it can even rain. On other occasions, it can drop to -10/-15 °C (5/14 °F), but it rarely drops below these values. This happens because of the influence of the ocean, which is reached by the Gulf Stream, and does not freeze even in winter. However, during the most intense cold waves from the North Pole, the temperature can drop further, as happened in January 1979, when it plunged to -26 °C (-15 °F).

In the short seasons of spring and autumn, ie in May and October, the temperature fluctuates around freezing, and snowfalls are frequent.

Summer, from June to September, is cold, with average temperatures a few degrees above freezing. On warmer days, highs can reach 10/12 °C (50/54 °F). The sky is often cloudy and the wind blows frequently, though less than in the rest of the year. The rains are frequent, though usually not abundant; in June and July, it rains on average for 8/9 days per month, and for 11 days in August. Sometimes it can snow even at sea level, especially in June and September, while on the slopes of Beerenberg, it normally snows even in summer.
In September, although the temperatures are still above freezing, the wind begins to blow with a higher intensity and the days get shorter in an evident manner.

Beerenberg, Jan Mayen

The amount of sunshine in Jan Mayen is definitely scarce, in fact, cloudy skies prevail throughout the year. However, there's a peak of 5 hours of sunshine per day in May and June, which drop to 4 hours in July and 3 in August: it's not much, since the day in this period is very long (it lasts 23/24 hours from May to July and 18 hours in August).
Jan Mayen - Sunshine

The temperature of the sea is always very cold, but not freezing, in fact it is about 1 °C (34 °F) from January to April, and 7 °C (44 °F) in August. Here are the average sea temperatures.
Jan Mayen - Sea temperature
Temp (°C)111123576432
Temp (°F)343434343637414543393736

Best Time

The best time to visit Jan Mayen is obviously the summer, especially from June to August, when the days are longer and the rains are less frequent. In June, the day lasts 24 hours, but the temperature is a bit lower.
However, you should bring warm clothes even in summer.