He discovered that his father had a special box in the basement

It sounds to me that the advice was one of style rather than grammar.

Many people think that that should not be overused, and that sentences flow better without it.

From the blog post "Overuse of That" by Billie Jo Schinnerer:

My finding is many times it can be deleted without being missed and often increases the flow of the passage. For example in the sentence below:

She found that she did not like the soup.
She found she did not like the soup.

This is, of course, a subjective opinion. Some people may find the first sentence preferable, others the second. (I personally prefer the second sentence—without that.)

No doubt the person who told you to not use that in your sentence simply feels that it sounds better without it. (In this case, my feeling is the opposite. I think your particular sentence sounds better with that.)

But different sentences have different cadences, and sometimes it might sound more natural to include that in one and exclude it from another. The context in which a sentence exists (its surrounding text) also makes a difference.

There are some legitimate reasons why that should not be removed.

In the blog post "When to Delete 'That'," Neal Whitman (a guest blogger for Mignon Fogarty) gives the following example:

Sometimes, omitting a "that" after a non-bridge verb goes beyond being slightly awkward and can actually be confusing. Here’s an example from Bryan Garner’s Modern American Usage:

• Son acknowledges being a member of a minority ... may have helped him turn his eyes abroad early.

The trouble here is that "acknowledge" can be a transitive verb. So when a noun phrase comes after it, such as "being a member of a minority," the reader might just take it as a direct object: “Son acknowledges being a member of a minority.” But whoops! The sentence keeps going, and the reader has to go back and reparse it. Garner calls this a miscue; sentences that produce miscues like this are called garden-path sentences.

He goes on to say the following:

Nouns that sound awkward if you delete a "that" include "fact." A phrase like "the fact Squiggly likes chocolate" is clear enough, but it’s really awkward-sounding.

He finally concludes:

If you’re a native English speaker, the main rule to follow here is to go by your ear. You probably know what sounds natural and what doesn’t, and all you need to do is give that native-speaker intuition more weight and authority than a rule stating that you should omit "that" whenever possible.

If you’re not a native speaker, I recommend keeping the "that" unless you’re dealing with a verb, noun, or adjective that you know will sound good without it. It’s safer to leave it in than to leave it out. As you write and read more, you’ll identify more of the words that allow you to omit "that."

Update: As was rightly pointed out in a comment, in the sentence in the question, discover is a transitive verb. Therefore, the sentence is also an example of a miscue or so-called garden-path sentence if that is removed. It would be more apparent if ellipses were used, but it's a good objective reason for why it sounds a bit awkward without it.