feat: Migrate from NextAuth to Better Auth

Backend changes:
- Add better-auth and pg packages
- Create Better Auth instance with PostgreSQL adapter
- Add Better Auth route handler at /api/auth/*
- Create migration script for Better Auth database schema
- Update main index to use Better Auth routes instead of custom auth
- Configure email/password and OAuth (GitHub/Google) providers

Frontend changes:
- Add better-auth client
- Create Better Auth client instance configuration
- Update lib/auth.ts to use Better Auth session
- Rewrite sign-in page with Better Auth methods
- Rewrite sign-up page with Better Auth methods
- Remove NextAuth route handler

Documentation:
- Add comprehensive migration guide with setup instructions
- Document environment variables and API endpoints
- Include testing checklist and rollback plan

Benefits:
- Unified authentication for both Elysia backend and Next.js frontend
- Database-backed sessions (more secure than JWT)
- Better TypeScript support
- Extensible plugin system for future features
- Active development and frequent updates

frontend/lib/better-auth.ts

@@ -0,0 +1,7 @@
+import { createAuthClient } from "better-auth/react";
+
+export const authClient = createAuthClient({
+  baseURL: process.env.NEXT_PUBLIC_API_URL || 'http://localhost:3001'
+});
+
+export const { signIn, signUp, signOut, useSession } = authClient;