Map from Google - Jan_Mayen

In 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, around freezing in May and October, while it rises a few degrees above in summer, between June and September.
Here are the average temperatures.
Average temperatures - Jan Mayen
Jan MayenJanFebMarAprMayJunJulAugSepOctNovDec
Jan MayenJFMAMJJASOND
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.
The volcano Beerenberg, 2,277 metres (7,470 feet) high, is covered with perennial snows already from an altitude of around a few hundred metres.
Precipitation amounts to about 700 millimetres (27.5 inches) per year, and is frequent and well distributed over the seasons; however, there's a relative maximun in autumn, and a relative minimum between April and July.
Here is the average precipitation.
Average precipitation - Jan Mayen
Jan MayenJanFebMarAprMayJunJulAugSepOctNovDecYear
Jan MayenJFMAMJJASONDY
Prec. (mm)555555404035456080806565675
Prec. (in)2.22.22.21.61.61.41.82.43.13.12.62.626.6
Days13111198891113151313134

Being above the Polar circle, in Jan Mayen you can see the midnight sun from late May to late July, and the white nights (when the sun falls below the horizon, but it doesn't get completely dark) from early May to mid-August.
In winter, the sun never rises from the 20th of November to the 20th of January, although around noon, when the sun is near the horizon, there's a bit of light also at the winter solstice (the shortest day of the year, the 21st of December).

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. In some periods, the temperature can rise slightly above freezing, up to 1/2 °C (34/36 °F), also 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, due to the influence of the ocean, which does not freeze even in winter, because of the Gulf Stream. However, during the most intense polar cold waves, the temperature can drop further, as happened in January 1979, when it plunged to -26 °C (-15 °F).

In the short 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 in the plains, 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.

The amount of sunshine in Jan Mayen is definitely scarce, given that cloudy skies prevail throughout the year; however, in May and June there are 5 hours of sunshine per day, which become 4 in July and 3 in August: it's not much, since the day in this period lasts 23/24 hours (which become 18 in August).

Beerenberg, Jan Mayen

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

When to go

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 have to bring warm clothes even in summer.