The Mysterious Island (novel)

The Mysterious Island (French: L'Île mystérieuse, 1874) is a novel by Jules Verne. I think it is clear from the book that the titular island is situated in the South Pacific, i.e. in Oceania with modern terms for the continents.
Elias Granqvist, 3 August 2004 & 28 November 2010

Quoting an English language translation at Project Gutenberg, Lincoln Island, as it was called, was at Longitude 150° 30′ west; latitude 34° 57′ south.
Peter Hans van den Muijzenberg, 28 November 2010

On Lincoln Island, castaways make do for three years on squalid conditions. Verne puts an emphasis on how these men managed to rebuild a civilized lifestyle from the ground up (something not less unlikely than his fanciful machines).
António Martins-Tuválkin, 20 February 2012

Flag of the United States

I have read a comic book version of this book. The castaways on the mysterious island are from the USA, and they produce an American flag for the boat they build. Only, the flag has one star too many; the extra star is for their island.
Elias Granqvist, 3 August 2004

In the novel, the fictional US flag with one star too many is described.

Quoted from Project Gutenberg:

Pencroff made a red, white, and blue flag, getting the dye from certain plants; but to the thirty-seven stars representing the thirty- seven States of the Union, the sailor added another star, the star of the "State of Lincoln:" as he considered his island as already annexed to the great republic.
"And," said he, "it is in spirit, if it is not in fact!"
For the present the flag was unfurled from the central window of Granite House and saluted with three cheers.

Elias Granqvist, 3 April 2005 & 28 November 2010

The text, in a different American version (at Project Gutenberg) is:

Pencroft also manufactured a flag, that flag so dear to every true American, containing the stars and stripes of their glorious Union. The colors for it were supplied from certain plants used in dyeing, and which were very abundant in the island; only to the thirty-seven stars, representing the thirty-seven States of the Union, which shine on the American flag, the sailor added a thirty-eighth, the star of "the State of Lincoln," for he considered his island as already united to the great republic. "And," said he, "it is so already in heart, if not in deed!"'

Peter Hans van den Muijzenberg, 21 February 2012

Positions of the stars

The graphical novel would probably be the only publication that shows positions for the stars, as I don't think the pattern is illustrated in the book. Or rather, it isn't illustrated that way: Though the text at some point says the flag is flying from their boat, the illustration shows a 4-5-4 star flag. Possibly that was intended as a boat ensign (though it would be a rather early example), or maybe it was drawn that way for the very same reason the boat ensign came into existence: Having so many stars would make them hard to see on such a small flag.

I expect Verne would really have thought of that flag as a flag with a minimal change to fit in the 38th star. I guess, the Army version of the 38-star Stars & Stripes, when it did come into being, would probably be the most likely design. Still, it would be fun to think up a different design a sailor on a far away island might have come up with. The book speaks of manufacturing a flag, colours, 38 stars and all. As such, the stars could be in any pattern, though it would not be a very extraordinary one, as the writer would have mentioned it if it were. So, staggered rows or spaced rows would be likely; a pattern outlining the island less so.
Peter Hans van den Muijzenberg, 20 & 21 February 2012

I'd say that a 4×5+6×3 staggered "round corner" pattern is the most likely.
António Martins-Tuválkin, 21 February 2012

I'd suggest he would stay close to the existing pattern. For 37 stars we show 8,7,7,7,8. If that was the most common design, than he might just have added a star to the middle row: 8,7,8,7,8.
Peter Hans van den Muijzenberg, 1 September 2012

L'Île Mystérieuse

Introduction

Flag of the United States

Positions of the stars

Suggested patterns of 38 stars