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

backend/src/routes/better-auth.ts

@@ -0,0 +1,7 @@
+import { Elysia } from 'elysia';
+import { auth } from '../auth';
+
+export const betterAuthRoutes = new Elysia()
+  .all('/api/auth/*', async ({ request }) => {
+    return auth.handler(request);
+  });