Sounds as though this only happens if they click the button before they fill
in the field, therefore you need to check for that case:

Private Sub Add_Click()
Dim DBS As Database, rs As Recordset

 If IsNull(Me.CName.Value) = True Then
   MsgBox "Enter a name first."
 Else
   Set DBS = OpenDatabase("\\Database Name")
   Set rs = DBS.OpenRecordset("tbl Customer Names", dbOpenTable)

   rs.AddNew
   rs![Name] = Me.CName.Value
   rs.Update

   DoCmd.Close acForm, "Customer Name Add"
 End If

End Sub

Personally, I'd be more inclined to use an INSERT INTO SQL statement rather
than bothering with a recordset:

Private Sub Add_Click()
Dim DBS As Database
Dim strSQL As String

 If IsNull(Me.CName.Value) = True Then
   MsgBox "Enter a name first."
 Else
   Set DBS = OpenDatabase("\\Database Name")
   strSQL= "INSERT INTO tblCustomerNames ([Name]) " & _
     "VALUES (""" & Me.CName.Value & """)"
   DBS.Execute strSQL, dbFailOnError

   DoCmd.Close acForm, "Customer Name Add"
 End If

End Sub

I'd also change the field name from Name. Name is a reserved word, and you
should never use reserved words for your own purposes. For a good discussion
on what names to avoid, see what Allen Browne has at
http://www.allenbrowne.com/AppIssueBadWord.html