The issue described above would occur if you are working in organisation mode and wanting to set a task to someone who is not the default contact for that organisation. Please see the example screenshot below showing contacts in organisation mode:
To resolve this you need to switch to individual person view of your contacts. This will show you individual people within organisations and not just company contacts.
To switch to the individual person view click on the person icon as shown below in red:
Your contacts will then be shown in person view: This will display both individual people contacts and company contacts.
Find the person to whom the task must relate to and open up their contact record.
When the contact record is opened up click on the Task icon and click Create New Task as shown below:
The task you create will automatically be associated to the individual person contact: