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/package.json

@@ -10,13 +10,14 @@
   "dependencies": {
     "@elysiajs/cors": "^1.4.1",
     "@elysiajs/swagger": "^1.3.1",
+    "bcryptjs": "^3.0.3",
+    "better-auth": "^1.4.18",
+    "dotenv": "^17.3.1",
     "elysia": "^1.4.25",
+    "jsonwebtoken": "^9.0.3",
     "pg": "^8.18.0",
     "postgres": "^3.4.8",
-    "dotenv": "^17.3.1",
-    "zod": "^4.3.6",
-    "bcryptjs": "^3.0.3",
-    "jsonwebtoken": "^9.0.3"
+    "zod": "^4.3.6"
   },
   "devDependencies": {
     "@types/pg": "^8.16.0",