Akiko has an aunt named Hana.

Literal

Akiko [to-には] Hana [quotative-という] name [possessive-の] aunt [subject-が] exist [polite-ます].

には is に + は — marking Akiko as the 'holder' of the aunt, topicalized. 'Akiko-AT-TOPIC, there is an aunt...' is the structure. ハナという名 uses the という pattern to attach a name to a noun: 'an aunt with the name Hana' = 'Hana called name of aunt.' The aunt is marked with が as the existing entity.