She´s having second thoughts about that job offer.-Which prize do you prefer: a blender or a tv set?
-A tv set, on second thoughts, a blender.
Is having second thoughts or on second thoughts related to a state of doubt? or sudden change of opinion or decision?
If you have second thoughts about something you change your opinion or start to have doubts about it.
There is a slight difference in usage though(not in meaning,mind you)
on second thoughts [BE]
on second thought [AE]
hope this helps.