The correct word here is 'give'.
If you don't 'give something a second thought' it means that you don't attach much significance to it. Note that it is always in the negative.
Don't confuse this expression with 'have second thoughts', in the plural, which means to begin to regret a decision you have made. For example, 'Mary said 'yes' when Peter asked her to marry him, but now she is having second thoughts. She's wondering whether he's the right man for her after all.'
As you can see, they are similar expressions with very different meanings.