What is the geochemical source for the helium detected in deep Arctic explosive eruptions?
Helium is formed through the radioactive decay of uranium 238, uranium 235  and Thorium 232. A helium atom is an alpha particle that has collected electrons. The only way that helium is produced on Earth is by such radioactive decay deep underground.

Generally, the helium travels along cracks until it reaches a subterranean cavern where it accumulates. 

Yellowstone National Park in the US is leaking helium. 

Quoting from the website http://www.yellowstonepark.com/2014/02/helium/


  the gas is gurgling up from a store that’s been trapped in the Earth’s core for up to 2 billion years. Scientists believe that the
  helium began its exodus up from the depths roughly 2 million years
  ago. 
  
  The helium was initially released when a “hotspot” of collected magma
  fought its way to the Earth’s surface and caused a series of volcanic
  eruptions, the most recent of which occurred 640,000 years ago.


The helium being release at Gakkel Ridge would be of a similar origin and it appears to be released in a manner similar to what is happening at Yellowstone NP. The article you provided a link for also states that carbon dioxide is being release at Gakkel Ridge, which is not inconsistent with the volcanic activity of the ridge.

From the website http://www.helium-corp.com/facts/heliumfound.html


  Carbon Dioxide is also found in some helium-bearing formations but the
  presence of this gas is primarily due to its proximity to ancient
  volcanic activity.


Other website that may be of interest are:
http://www.pa.msu.edu/sciencet/ask_st/051095.html 

http://www.anl.gov/articles/underground-helium-travels-earth-s-surface-aquifers-new-study-says