I seem to recall that it is not actually possible to ask a cell  whether it
has a name. There are a couple of ways to work around this.

1) Assign the name of the cell to a string, but: (a) prefill the string
with a special value or an empty string, and (b) put the assignment
statement under On Error Resume Next. E.g.

Option Explicit

Private Sub Worksheet_Change(ByVal Target As Range)
If Target.Address <> "$A$1" Then
Dim XX As String
XX = "No"
On Error Resume Next
XX = Target.Name.Name
On Error GoTo 0
Range("A1").Value = XX
End If
End Sub

puts into A1 the name of the cell if it has one, or "No".

2) Use Intersect. E.g.

Private Sub Worksheet_Change(ByVal Target As Range)
If Not Intersect(Target, Range("aa")) Is Nothing Then
Range("A1").Value = "aa"
ElseIf Not Intersect(Target, Range("bb")) Is Nothing Then
Range("A1").Value = "bb"
ElseIf Target.Address <> "$A$1" Then
Range("A1").Value = "Nothing"
End If
End Sub

does specific  checks for names "aa" and "bb".