The VBA script below saves a sheet to xlsx file with today's date in the name.

 Use the GetSaveAsFilename function:

' Source:
' Description: Saving Sheet in .xlsx File

Sub SaveSheet()
    On Error Resume Next
    Const REPORTS_FOLDER = "My Reports\"                   'Folder to save our file
    MkDir ThisWorkbook.Path & "\" & REPORTS_FOLDER
    ChDrive Left(ThisWorkbook.Path, 1): ChDir ThisWorkbook.Path & "\" & REPORTS_FOLDER
    TbName = "MyFilename" & Format(Date, "YYYYMMDD") 'FileName plus today's date
    filename = Application.GetSaveAsFilename(TbName, "Excel (*.xlsx),", , "Name", "Save")
    If VarType(filename) = vbBoolean Then Exit Sub
    Err.Clear: ActiveSheet.Copy: DoEvents
    If ActiveWorkbook.Worksheets.Count = 1 And ActiveWorkbook.Path = "" Then
        ActiveSheet.Buttons.Delete                          'deleting buttons
        ActiveSheet.Range("A:Z").Value = Range("A:Z").Value 'cleaning formulas
        ActiveWorkbook.SaveAs filename, FileFormat:=51      'setting .xlsx format
        ActiveWorkbook.Close False
    End If

End Sub


Download an example xlsm