This Publisher code is annoying to debug. Try this one (again, after selecting the frame where the name should sit). It works for me and appears a bit more robust.
Code:
Sub TestExport()
Dim sName As String, sPath As String, aRng As TextRange
Set aRng = Selection.ShapeRange(1).TextFrame.TextRange
With ActiveDocument.MailMerge.DataSource
.ActiveRecord = 1
Do
sName = .DataFields.Item("Name").Value
aRng.Text = sName
'sPath = ActiveDocument.Path & "\" & .DataFields.Item("JpgFileName").Value
sPath = .DataFields.Item("JpgFolderPath").Value & "\" & .DataFields.Item("JpgFileName").Value
ActiveDocument.Pages(1).SaveAsPicture Filename:=sPath, pbResolution:=pbPictureResolutionWeb_96dpi
If .ActiveRecord = .RecordCount Then Exit Do
.ActiveRecord = .ActiveRecord + 1
Loop
End With
End Sub