I want to somehow designate some of regular body text to be picked up in a cross-reference. How do I do that?

In an unstructured document:

• You can select part of a paragraph as the cross-reference text by setting it to be a run-in paragraph, and using the <$paratext> building block. Format any preceding text as another run-in paragraph, and any following text as a regular paragraph.
• You can use arbitrary words as the cross-reference text by creating invisible text in an anchored frame outside the main flow, and using the <$paratext> building block. See Problems with spot cross-references.

In a strucured document:

• You can select part of a paragraph as the cross-reference text by wrapping it in an element, and using the <$elemtext> building block.
• You can use arbitrary words as the cross-reference text by setting an attribute to the words (e.g. xreflabel in DocBook), and using the <$attribute> building block.

See also:
Problems with spot cross-references

In unstructured mode, all you can do is pick up the entire content available in the <$paratext> building block. There is no marker text available.

You can kludge this by creating an anchored frame with a text frame that holds the desired text of the cross-ref that must use a unique paratag that has a colour attribute that is set to be invisible in the Color View mode used during the publication step. [Set the Aframe to outside the column  or text frame so that it doesn't  visually affect the appearance of your content flow.]

In structured mode, IIRC, you need to wrap the desired text in a separate element.