Adding a User to a Distribution Group

In Active Directory, locate the user and open Properties. In the Member Of tab, "Add" the distribution group to which they should belong. Hit Apply to save the change.

To give the user permissions to send e-mails as this distribution group, log in to the Exchange Admin Center. Under "groups," locate and open the distribution group. Under the "group delegation" tab, in the "Send as" section, click the '+' sign to add the user. Hit "save" to apply the change.
If you receive an "Access denied - insufficient access rights" error (see screenshot), proceed to Steps 3 and 4.

To bypass this error and add the user to the Send As list, go into the properties of the user in Active Directory. In the Security* tab, click "Advanced" and check the box that says "Include inheritable permissions from this object's parent." Click Apply and OK.

(*If you do not have a "Security" tab, go to the main Active Directory window, View menu --> Advanced Features.)

In Active Directory, navigate to the properties of the distribution group. Click on the Security tab. Add the user to that list, and check the boxes to Allow: Read, Send As, Read Exchange Information, Read Exchange Personal Information, and Read phone and mail options (5 boxes checked). Click OK.
Try adding the user to the Send As list again in Exchange Admin Center.