Feature Request: Allow customizing VSCode tab label for Claude Code webview

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Since Claude Code tabs are webviews, third-party extensions like "Rename Tabs" cannot rename them — the VSCode API restricts tab label modification to the owning extension only. There is currently no way for users to differentiate between multiple Claude Code tabs.

Proposed Solution

Allow the tab label to be customized in one or more of the following ways:

1. Use the project/workspace name as part of the tab label automatically (e.g., Claude Code — my-project)
2. Use a snippet of the conversation title or first message as the tab label
3. Allow the user to manually rename a Claude Code tab via a command palette option or right-click context menu

Alternative Solutions

Allow the tab label to be customized in one or more of the following ways:

1. Use the project/workspace name as part of the tab label automatically (e.g., Claude Code — my-project)
2. Use a snippet of the conversation title or first message as the tab label
3. Allow the user to manually rename a Claude Code tab via a command palette option or right-click context menu

Priority

Critical - Blocking my work

Feature Category

CLI commands and flags

Use Case Example

Allow the tab label to be customized in one or more of the following ways:

1. Use the project/workspace name as part of the tab label automatically (e.g., Claude Code — my-project)
2. Use a snippet of the conversation title or first message as the tab label
3. Allow the user to manually rename a Claude Code tab via a command palette option or right-click context menu

Additional Context

Users with multiple Claude Code tabs open should be able to identify each session at a glance, similar to how browser tabs show page titles.
All Claude Code tabs show the same label, making multi-session workflows confusing.