They (the linguists who have studied this) seem to believe that the word Bodø is only a variant of the old local name Bodin, that is Bodhvin in the old language. They are quite sure that the last part is "vin" (they can be sure while this is a frequently found part of many very old norwegian place names), which means (or was in old norwegain the word for) "natural meadow" (that's also what the vikings called the american coast; Vinland). They also think first part might be oldnorse bodhi, (modern norw. "båe" or "grunnskjær") a rock that doesnt really stick up of the water, you can only see it from the waves/water breaking around/over it. So with rocks and green, they are not that far out. But the "islands" is an old linguistic misunderstanding. "Ø" is the danish word for norwegian "øy", which means "island". And since we wrote Danish for quite a long period (until less than a hundred years ago), an old "jump-to-conclusions-theory" has been that "ø" means "island". But it doesn't fit the landscape very well, and it doesn't fit linguistically, so that theory is abandoned. (The japanese seems to have recycled an out of date-interpretation.)
I used to live in Bodø, quite many years, and I think "the meadow by the rocks that are washed by the sea" (or somthing like that) fits the landscape very well, and it fits the linguistic interpretation. (There isn't much left of the meadow, of course but some places one can still see it.)
My source is "Norsk stadnamnleksikon", Samlaget 1997 (and my perceptions of the place and the little knowledge from the place name studies that was part of my education ;-)