[AI Chat] Prepare new NTP search box for day zero experiment (uplift to 1.93.x)

[brave-builds]

Uplift of #38638
Resolves brave/brave-browser#57701

Pre-approval checklist:

• You have tested your change on Nightly.
• This contains text which needs to be translated.
  • There are more than 7 days before the release.
  • I've notified folks in #l10n on Slack that translations are needed.
• The PR milestones match the branch they are landing to.

Pre-merge checklist:

• You have checked CI and the builds, lint, and tests all pass or are not related to your PR.

Post-merge checklist:

• The associated issue milestone is set to the smallest version that the changes is landed on.