Hello,

If your family income is actually less than Rs. 1 lakh, then you must correct it during the JEE correction window .
If you keep the wrong income (up to Rs. 1 lakh), the college may not give you fee waiver or EWS-based tuition fee benefits. So correcting it is better.

About the correction fee:
Usually, JEE does not charge extra for changing basic details like income or category documents during the correction window. But if you change something that affects the application fee (like category), then a fee difference may apply.
For income detail change only, you normally do not need to pay anything .

So yes, correct it if your real income is below Rs. 1 lakh. It will help you during counselling and fee waiver.

Hope it helps !