[FEATURE] Session naming: consistent rename across all clients

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

Session renaming is inconsistent across clients:

• /rename exists in the terminal TUI but is not available in the VSCode extension or in-chat
• Renaming from the mobile remote interface doesn't propagate to other clients

Desired: a single session name, settable from any client (terminal, VSCode, mobile, in-chat), stored server-side and reflected everywhere that session appears.

Proposed Solution

• In-chat: a /rename <name> command available in all clients (terminal, VSCode extension, mobile)
• VSCode: right-click a session tab to rename the current session, or right-click an entry in the session list to rename any session
• Mobile: rename already works — just needs to sync
• On launch: claude --name <name> to set the name upfront (already exists for --bg; extend to foreground sessions)

The name should be stored server-side so it appears consistently wherever the session is listed — resume list, VSCode sidebar, mobile session list.

Alternative Solutions

None found. I just get annoyed.

Priority

Medium - Would be very helpful

Feature Category

Interactive mode (TUI)

Use Case Example

I work on multiple Perl modules simultaneously. I open a Claude Code session in VSCode for Devel::Bug, another for Data::Trek, and a third for a backup utility. Without named sessions, the resume list and session history show only timestamps or auto-generated titles — I have to open each one to figure out which is which.

With this feature:

1. I start a session in the Devel::Bug project directory and type /rename Devel::Bug
2. That name appears in the VSCode session list, the terminal resume list, and on my phone
3. Later I pick up the session on mobile — it's already named, no action needed
4. A week later I use /resume in the terminal — I can immediately identify and reopen the right session by name

As an enhancement, the VSCode extension could automatically name new sessions after the project directory (or workspace name) when a session is first opened, eliminating the need to rename manually in the common case.

Additional Context

_No response_