In Iceland only names which appear on the Personal Names Register are allowed to use. Other names cannot be used, but it is possible to apply to a committee for permission to use a name which is not yet listed.
The committee does not accept every name. A name submitted to the naming committee for approval is considered for its compatibility with Icelandic tradition and for the likelihood that it might cause the bearer embarrassment. It must be compatible with Icelandic grammar and contain only letters occurring in the Icelandic alphabet. A name's grammatical gender must match the sex of the person bearing the name. There are occasional exceptions, e.g. if a name has traditionally been used by a certain number of Icelanders.
If a name is rejected it will not be added to the list of approved Icelandic names.
Source: Mannanafnaskrá at island.is (as of February 2019)