To have s.o. weave a mat (using the materials of the one having a mat woven); to weave a mat for s.o. (using materials belonging to the one hiring to have a mat woven).
I'm going to have s.o. weave my pandanus strips into a mat because I can't keep up with all the weaving I have to do.
Have that woman weave for you if she doesn't have any more pandanus strips.
Who wove your pandanus strips for you?
Weave that woman's pandanus strips for her.