Which of two conflicting abilities takes precedence?
Summary
The player whose turn it is takes precedence. In resolving their ability, the current players makes it impossible for the other player to resolve their ability, so the other player can't trigger it.
In the Runner's turn the Runner chooses, and the Corp does nothing.
In the Corp's turn the Corp looks at the Runner's grip and chooses.

Details
I will be referring to these rules. From page 21:

Triggered Abilities
In order to use a triggered ability a prerequisite must be met. This prerequisite is either a trigger cost that must be paid (PAID ABILITY) or a trigger condition that must be met (CONDITIONAL ABILITY). Once an ability is triggered, its effect is resolved immediately and can only be stopped by PREVENT
or AVOID effects. Players must follow all restrictions on the cards when triggering abilities.

From page 22:

Simultaneous Effects
When one or more abilities have the same timing trigger or can be triggered at the same time, each player chooses the order his own abilities trigger. A player can trigger an optional conditional ability before a required conditional ability if they both have the same trigger condition.
If players ever want to perform simultaneous effects at the same
time, the player whose turn it is resolves all of his effects first.

Both of the cards in question have conditional triggered abilities. For Titanium Ribs, this is &quot;you take damage&quot;. For &quot;Chronos Protocol&quot;, this is &quot;the first time this turn the Runner takes net damage&quot; (note that this is an optional ability).
The first time the Runner takes net damage each turn, both of these conditional abilities can trigger. If they do (the Corp chooses to trigger the optional ability), we must follow the rule for simultaneous effects (the player whose turn it is resolves all of his effects first), while keeping in mind the rule on triggered abilities (once an ability is triggered, its effect is resolved immediately).
Additionally, from the rules FAQ, page 3:

A player cannot trigger an action/ability unless he is also able to resolve it.

If one player has resolved his triggered ability and trashed a card, the other player will not be able to resolve their own ability, and so cannot trigger it.
Resolution during the Runner's Turn

The Runner resolves all of his effects first:

Titanium Ribs:

The Runner chooses the card to trash.
The Runner trashes that card.




The Corp resolves all of his effects second:

Chronos Protocol:

In order to resolve Chronos Protocol, the Runner must trash a card due to damage.
The Runner has already trashed a card for this damage, so Chronos Protocol could not be resolved in this situation.
Because of this, the Corp cannot trigger Chronos Protocol.





Resolution during the Corp's Turn

The Corp resolves all of his effects first:

Chronos Protocol:

The Corp looks at the Runner's grip and selects a card to trash.
The Runner trashes that card.




The Runner resolves all of his effects second:

Titanium Ribs:

In order to resolve Titanium Ribs, the Runner must trash a card due to damage.
The Runner has already trashed a card for this damage, so Titanium Ribs could not be resolved in this situation.
Because of this, the Runner cannot trigger Titanium Ribs.