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I apologise for any mistake, but this idiom sounds too modern to be explained by etymology.

I want to dredge below the definition, the first one of which is too violent and doesn't explain the neutral meaning of the second here:

2. Have as the main topic or point of interest:

Source: pp 205, Thinking like a Lawyer, Frederick Schauer

But given that questions of law almost always turn on determinations of fact, and given that determinations of fact are in numerous ways structured by legal rules and by characteristic ways of reasoning, to exclude questions of fact from the topic of legal reasoning seems peculiar.

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  • This is definitely a question of etymology, even if it is a recent development. I recommend asking on ELU.
    – Tyler James Young
    Commented Aug 25, 2014 at 15:59

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In the OED, the definition to which you refer is written thus:

3. turn on or upon (fig.):

a. To hinge upon, depend on, have as the centre or pivot of movement or action.

b. To have as its subject, be about or concerned with, relate to: usually said of conversation or debate.
Definition of “turn” via oed.com (login required)

3.a. appears to be the prior sense of the word, and the earliest recorded usage is in 1661:

They that turn upon this hinge, I mean that receive Procurations upon the ground of Custome.
Source: J. Stephens Hist. Disc. Procur. 26, via oed.com (login required)

The note for 3.b. (the definition you’re interested in) says “App[arently] orig[inally] a development of prec[eding] sense, but often associated with other senses: cf. 28.”

The cross-referenced definition is:

28. a. intr. To direct one's mind, desire, or will to or from some person, thing, or action.

Based on all of that, I conclude that this use of “turn on” to mean “have as the main topic” is based on the metaphor of a hinge. This relatively small, important apparatus is at the center of a larger object’s arc of motion. The presence of other senses of “turn”, like that of changing focus to something, probably influenced this transition and left us in the present state on not even needing to refer to a hinge directly.

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This use of "turn on" means "depend on".

I am most familiar with this use of "turn on" in two situations:

  • In law, examples like yours: "The legal case turned on <this critical issue>." These examples seem to be battle metaphors, as explained below.
  • In military history, when an author is explaining how a battle or war turned out. "The battle turned on <this critical event>." Often, the author is using the word "turned" or "pivoted" literally. In many battles, the armies spread out into lines that face each other. If one part of one army is beaten, the rest of that army will tend to fold up -- either falling away from the hole in its lines, or collapsing into it. Either way, the battle lines will tend to rotate.

Here is a great passage that uses the military history meaning of "turn on" metaphorically. The passage is from Eric Flint's alternate history book 1632, but his description of the Battle of Breitenfeld is according to the real history. (The Swedish army's pivoting maneuvers were critical to how the battle turned out.)

Breitenfeld.

All the legends revolve around that place. They pivot on that day. Wheeling like birds above the flat plains north of Leipzig on September 17, 1631, they try to find sharp truth in murky reality. Never seeing it, but knowing it is there.

The legends would be advanced, and refuted, and advanced again, and refuted again—and it mattered not in the least. Breitenfeld remained. Always Breitenfeld.

After Breitenfeld, how could the legends not be true?

As you probably know, "turn on" has some other meanings. The most common are:

  • To start something, such as an engine or a person's sex drive. My mental picture is of a person turning a key to start an engine.
  • To betray someone. My mental picture is of a person physically turning to stab a companion.
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