Because costed abilities are used during step 3.1 of a run ("Icebreakers can interact with the ice"), if the Corp uses Tyr's Hand to prevent a subroutine from being broken, the Runner can just pay to break it again (if they can afford to), correct?

Also, is the "buyout" on the Bioroid ice considered a costed ability? When exactly does that happen? Step 3.1?

Effects that "prevent" something else have a special timing, as explained in the rulebook. They happen just before the effect they're preventing resolves but after it's paid (Decoy is such an effect). And yes, nothing in Tyr's Hand prevents the runner from paying to break the subroutine again (because it's still his turn to use paid abilities).

The biggest thing preventing the runner from re-breaking a subroutine prevented with Tyr's Hand is a lack of clicks. If they're paying 1 credit to break a routine, it's not a great use just to force them to pay 1 more. But if they're running on click 3 and counting on E3 implants to get them past your Janus, suddenly they can't break ANY of his routines after you counter the first "click to break." It's situational, but that's why it only costs 1 credit.

The runner is free to break again. The jest of Tyr's Hand is to negate one break when the runner spends clicks to break subroutines, not when the runner spends creds to break routines.

It's great against runners trying to break high-strength Bioriods with clicks (also helps shut down E3 in certain situations as outlined above).

Just realized: I guess it also screws over the first function of Deus X.

Heh, that would be a lovely play. Espeically if you're using your Deus Ex to get out of a chum, too. I wasn't planning on buying the expansion right away, but the more cards get spoiled for it, the more that resolution erodes.

I don't think the runner can break the subroutine again even he's afford to do so.

I think would function just like the FAQ: Trash prevention effects (just like sacrificial construct) can prevent a chosen program from being trashed, regardless of how many times was chosen.

This answer is about Aggressive Secretary, that is when the same program is chosen multiple times by a single effect. To prevent multiple effects from trashing the program (Archer, Ichi 1.0), you'd have to use multiple prevention effects.

Oh, that clarifies it.

The only bit I don't understand is why a runner would ever be relying on clicks to get past ice, thus making Tyr's hand a great card to use. Have I totally misunderstood some types of ICE, are there cases where a runner MUST use clicks to break ice subroutines?

There is no ice (as of yet) where the runner must use clicks to break through bioroid ice. But very often runners rely on clicks to break some ice either because they don't have the required breakers yet, or if it's to expensive to get up to strength of the ice in question. It's actually quite common in both the early game and the late game when you're looking for that last agenda and you don't care about breaking every subroutine on Janus or Heimdall.