Rules FAQ pt 1

What happens if my opponent chose military claim for Calm Over Westeros? They reduced your claim, but they don't prevent it from happening.

Here you replace the 'normal' claim (of killing 0 characters) with killing the character you chose. Note that the opposite applies too - if you have a claim 2 (or higher) plot and choose to use Seastone Chair, you'll be replacing the 'normal' claim (of killing 2+ characters) with killing only the one you chose.

Rules FAQ pt 2

From RRG pg 18: A replacement effect is an effect (usually an interrupt) that replaces the resolution of a triggering condition with a different means of resolving that same triggering condition, but in such a manner that the triggering condition is still considered to occur. The word 'instead' is frequently indicative of such an effect. After all interrupts to the triggering condition have resolved and it is time to resolve the triggering condition itself, the replacement effect resolves instead.

Claim replacement effects are triggered during the C step of DUCK (after Determining the winner and gaining the Unopposed bonus but before Claim is applied). These effects do not resolve immediately, rather they modify the way claim will be applied after all interrupts are played. For example, with Seastone Chair you choose a character, and set up an effect that will kill that character when claim resolves. But after you trigger Seastone Chair, and before you resolve claim, your opponent might play his own interrupt to applying claim (like Vengeance of Elia) and set up a new effect that will modify the way in which claim will resolve. This new replacement effect will override the previous one. Once all players are done playing Interrupts to "when claim is applied", resolve the claim in accordance with the most recently played replacement effect.