[FEATURE] Allow renaming conversation tabs in VS Code extension

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

When working on multiple tasks in the VS Code extension, conversation tabs are auto-titled and cannot be renamed. The CLI supports /rename and claude -n, but these commands don't work in VS Code. With several tabs open, it becomes difficult to distinguish between conversations, especially when switching back to a previous one.

Proposed Solution

Add the ability to rename conversation tabs in the VS Code extension. Options:

Right-click a tab → "Rename"
Support /rename <name> within VS Code conversations
Editable tab title on double-click
The name should persist when the conversation is resumed.

Alternative Solutions

Currently there is no workaround — tabs cannot be renamed or named at creation in VS Code. The only option is to close and mentally track which tab is which by scrolling through conversation history.

Priority

Medium - Would be very helpful

Feature Category

CLI commands and flags

Use Case Example

I'm working on multiple independent tasks (e.g., fixing a bug, adding a new feature, refactoring code)
I open separate Claude Code tabs for each task
All tabs have auto-generated names that don't reflect the task
When I switch between tabs, I have to scroll through each conversation to remember which is which
With this feature, I could name them "bug-fix", "new-feature", "refactor" and instantly find the right one

Additional Context

_No response_