I had the same problem, hopefully your problem is occurring for the same reason, and the following solution will help you.
Step 1: Make sure you have set up iCloud for your iPhone, which I am sure you have.
Step 2: Go to Settings>iCloud and turn off Contacts. If you are experiencing the same problem I had (i.e. no iCloud contacts set up) it will not matter if you delete or save when prompted, that is up to you.
Step 3: Delete the current contact that is assigned to your iPhone/Siri (this will also remove it under "My Info" in the Siri settings).
Step 4: Go to iCloud.com on a computer, log in, and select Contacts. Now press the + button on the bottom right of the first "page," and create a contact with your name/information.
Step 5: Once created, select your contact, press the Settings button on the bottom left, and choose Make this my card.
Step 6: On your iPhone again, return to Settings>iCloud and turn Contacts back on, selecting merge when prompted.
Step 7: Go to Settings>Mail, Contacts, Calenders; scroll down to the section under the Contacts heading, select Default Account and choose iCloud.
Step 8: Return to your contacts.
a) if your name is already there (matching the one you created at iCloud.com). Select it and look under the Linked Cards heading, to make sure that there is a link to your iCloud contact. Also, check under Settings>General>Siri to make sure that it has the the contact in the My info box.
b) if it is not there, create a new contact, and enter your first and last name exactly as it is entered in iCloud (it is not necessary to provide information beyond your first and last name). Now press done, and you should see your iCloud account appear below the Linked Cards heading. Select the iCloud account and you can make changes to the contact from your phone, including adding relationships that will work with Siri (e.g. Mother, Father, Brother, etc.). You should also be able to use Siri as you have tried in the past, but only got the "I can't add that relationship to this account" response.
Optional Step 9: If it still does not work, you may want to try using this process to add your important "relations" contacts to your Contacts List via iCloud.com, and adding the relationships to you contact on iCloud.com as well.
I hope this helps! Good luck!