To be able to achieve this, we need to write some code in the Worksheet module that will trigger an action when any change happens.

Here's the code to be written in Worksheet module:
Option Explicit
Public CellCheck As Boolean

Private Sub Worksheet_Change(ByVal Target As Excel.Range)

    If Range("A1") = "1" And CellCheck = False Then
        Call StartBlink
        CellCheck = True
    ElseIf Range("A1") <> "1" And CellCheck = True Then
        Call StopBlink
        CellCheck = False
    End If

End Sub
Here's the code that is to be written in a Module:
Option Explicit
Public RunWhen As Double

Sub StartBlink()

    If Range("A1").Interior.ColorIndex = 3 Then
        Range("A1").Interior.ColorIndex = 6
        Range("A1").Interior.ColorIndex = 3
    End If
    RunWhen = Now + TimeSerial(0, 0, 1)
    Application.OnTime RunWhen, "StartBlink", , True

End Sub

Sub StopBlink()
    Range("A1").Interior.ColorIndex = xlAutomatic
    Application.OnTime RunWhen, "StartBlink", , False

End Sub
You may put two buttons on the worksheet - one for starting the blinking process and one for stopping it. Once the button is inserted, simply right click on it and click on 'Assign Macro' and select the appropriate macro.

I'm also attaching the worksheet.