How to make part of one image the entire background of another in Photoshop?

Make a selection around the part of the pattern you would like to use as a fill pattern. Select Edit > Define New > Pattern.

Then make a selection for the area you want to fill. If it’s black it might be easiest to use the magic wand tool to select the area. Or select the black area then invert the selection (Select > Inverse) to select everything which is not black.

Make the selection into a layer mask. I like to do this by making a selection and then clicking the Quick Mask button at the bottom of the layers pallet.

Make sure your new layer is under your original layer. Then Fill the new layer with a pattern. Edit > Fill. Select Pattern from the drop down and then click the pattern you defined earlier.