Custodial parents, residing in Iceland and have an unmarried minor (under the age of 18) in their care, may apply for a residence permit for the minor provided at least one parent is:
an Icelandic citizen
other Nordic citizen (see note below about EEA/EFTA citizens*)
a foreign national with permanent residence in Iceland
a foreign national whose temporary residence in Iceland is based on:
work as a qualified specialist
work as an athlete
marriage or cohabitation
studying as a graduate or post-graduate student
grounds of international protection
grounds of humanitarian views
grounds of special ties to Iceland
In order to apply for a minor, that minor must be in your custody and financially supported by you. Furthermore, the minor must live with you if the permit is approved. Note that in cases of joint custody, both custodial parents must sign the form.
The Directorate of Immigration provides the same form (common application) to apply for residence permits for children under 18, volunteers, missionaries, people with legitimate and special purposes and people with special ties to Iceland. So be aware that not all fields on the form will apply to you.
For more details on this kind of permit, please see the Directorate of Immigration.
*Note that a citizen of an EEA/EFTA state may sponsor their minor children from outside the EEA/EFTA for a residence card instead of a residence permit.