Conditional + Let you know

Solution 1:

The second sentence is definitely the correct one:

If I change my mind, I'll let you know.

You or your friends are probably confused because when "I'll let" is spoken, all those "L"s blend together and sound like "I let". Don't be fooled by that.

Try the same tense with different words. Which sound correct to you?

Incorrect: If I change my mind, I buy the product.
Correct: If I change my mind, I'll buy the product.
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Incorrect: If I win a lottery, I donate the money.
Correct: If I win a lottery, I'll donate the money.

Solution 2:

The first one is a continuous tense, indicating that any time you change your mind, you already do - and will continue to - let them know. This is probably not what is intended.

The second and correct one means that in the case where you change your mind, you will at that point, let them know. In fact, the context is also important, because it means that if you change your mind about this particular issue, you will inform them. In the first case, because this is a continuous use, the context is not necessarily implied.