Fix profile name update persistence

Replace placeholder implementation with proper API integration:
- Create /api/user/profile PUT endpoint with JWT validation
- Update profile page to call actual API instead of setTimeout
- Use refreshUser() to update UI immediately after changes
- Ensure name changes persist to database and across page refreshes

🤖 Generated with [Claude Code](https://claude.ai/code)

Co-Authored-By: Claude <noreply@anthropic.com>

app/[locale]/profile/page.tsx

@@ -28,7 +28,7 @@ import {
 } from '@mui/icons-material'
 
 export default function ProfilePage() {
-  const { user } = useAuth()
+  const { user, refreshUser } = useAuth()
   const t = useTranslations('profile')
   const [isEditing, setIsEditing] = useState(false)
   const [name, setName] = useState(user?.name || '')
@@ -40,10 +40,31 @@ export default function ProfilePage() {
     setMessage('')
 
     try {
-      // TODO: Implement profile update API
-      await new Promise(resolve => setTimeout(resolve, 1000)) // Placeholder
-      setMessage(t('profileUpdated'))
-      setIsEditing(false)
+      const token = localStorage.getItem('authToken')
+      if (!token) {
+        setMessage(t('updateError'))
+        return
+      }
+
+      const response = await fetch(`/api/user/profile?locale=${navigator.language.split('-')[0]}`, {
+        method: 'PUT',
+        headers: {
+          'Content-Type': 'application/json',
+          'Authorization': `Bearer ${token}`
+        },
+        body: JSON.stringify({ name })
+      })
+
+      const data = await response.json()
+
+      if (response.ok) {
+        setMessage(t('profileUpdated'))
+        setIsEditing(false)
+        // Refresh user data in context to show updated name
+        await refreshUser()
+      } else {
+        setMessage(data.error || t('updateError'))
+      }
     } catch (error) {
       setMessage(t('updateError'))
     } finally {