The proper name is Justøya, which is basically "The Justøy" and the name generally used when referring to the island. The word Justøy will be found in names like "Justøybroa" - "The Justøy Bridge" og "Justøyveien" - "The Justøy Road" because the definite article -a or -en (equivalent to "the") moves to the last word in the compound word.
That said, some places it is referred to as simply Justøy, among other places in the wikipedia article about it, so I think this warrants adding it as an alternative name.