Is there an "official" name for a heading prefix, or qualifier?

Solution 1:

According to Dictionary of Layout & Typography Terms, the element containing the text "Challenge #1" in your example is variously known as a kicker, an overline, or an eyebrow:

KICKER Short, (sometimes underlined) phrase introducing a headline. Also called overline or eyebrow.

But elsewhere in publishing, I have heard the term kicker used to refer to the opening of a story when it is set in all-caps or boldface type or both, as the entry for the term in Glossary of Magazine and Newspaper Terms indicates:

Kicker - The first sentence or first few words of a story's lead, set in a font size larger than the body text of the story.

Yet another term for the element you are asking about is strap. From Glossary of Newspaper Terms:

Strap subsidiary smaller headline placed over a main headline

At the computer magazines where I worked for many years, we called text elements of this type eyebrows—a designation that always seemed very appropriate to me because, in our layout design, the text appeared in small all-caps drop-out letters in a narrow bar of color above the headline. Graphically, the element really was reminiscent an eyebrow.