For convenience, I reproduce the relevant text from the FAQ:

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(3.8) Gaining Abilities
If any card effect has the text "character
gains...," any ability (traits, keywords, icon
modifiers, or STR modifiers are not considered
abilities) following the word "gains" is
considered to now be a character ability.

However, that effect is not assumed to be a
part of that character's text box. Any effect that
blanks a card's text box is assumed to blank
that card's printed text box.
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Let's take as an example "Kennel Master". Assume his controller controls 3 or more direwolf cards, resulting in him gaining an ability. Do I understand correctly, then, that if one blanked the text of "Kennel Master", but the condition for giving him his response ability still held, he would still retain his response ability?

He would not, since the text causing him to gain that ability is also part of his text box. Here's an example:

Tyene Sand says "All Sand Snake characters gain stealth." If you blank Obara Sand (who does not normally have stealth), she would still have stealth, since Tyene is still hanging out, giving it to her.

However, blank Tyene, and suddenly there is no effect that gives all Sand Snakes stealth, and so Obara would lose stealth.

JackT said:

However, that rule of thumb only works when effect A is on a different card from card X. That's because effect A still can be blanked, cutting off access to effect B. So when effects A and B are on the same card - like with Kennel Master - blanking the card will eliminate the use of the gained effect; not because the gained effect is blank, but because the effect giving the gained effect is blank.

It would be different if the ability was gained through a triggered or passive effect rather than a constant effect (like Kennel Master). If you blank LoW Robb Stark after he has used his ability, characters with the named trait would still have their STR boosted, because the effect has resolved.