The central issue is the same as the Donofrio case: whether Barack Obama is not eligible to the office of President due to the fact that he was a British citizen at the time of his birth.
Article 2, Section 1, of the U.S. Constitution specifically states: "No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President."
His father's nationality, as a British citizen, appears to make Barack Obama ineligible for the presidency as he is not a natural born Citizen.
"When Barack Obama Jr. was born on Aug. 4, 1961, in Honolulu, Kenya was a British colony, still part of the United Kingdom's dwindling empire. As a Kenyan native, Barack Obama Sr. was a British subject whose citizenship status was governed by The British Nationality Act of 1948. That same act governed the status of Obama Sr.'s children."
Cf. http://www.fastcheck.org/
Of relevance is the newly discovered ineligibility of 21st President Chester Arthur due to his having been born as a British subject. This is relevant to the case at hand in that Justice Gray--who wrote the seminal opinion in United States v. Wong Kim Arc-- was appointed by Chester Arthur.
The Justice who wrote the opinion was appointed by Chester Arthur.