A preposition before a that-clause is dropped. So it can only be

1 No one had told Smith (that) there would be a lecture the following day

Or

2 No one had told Smith about there being a lecture the following day.

The normal formulation would be 1. 2 is a bit stilted.

It's so much easier to see this when you fill in the blank.

A) No one had told Smith about there being a lecture the following day

This reads fine to me.

B) No one had told Smith about there would be a lecture the following day

Adding would removes the need for the ing on being. This change is fine.

To me B is screaming for one more word:

C) No one had told Smith about how there would be a lecture the following day

Sure C is full of froth and bubble but it doesn't set my teeth on edge like B does.

B also can get by with one less word

D) No one had told Smith about there would be a lecture the following day

Short answer

The word about cannot take finite clauses as a Complement. A finite clause is a clause we can use like a sentence on its own. The word about CAN take -ing clauses as a Complement:

• *No one had told Smith about [there would be a lecture the following day]. (ungrammatical)

• No one had told Smith about [there being a lecture the following day].

For this reason the correct answer is (B).

Full answer

The verb TELL takes several different patterns. The verb TELL can take the following types of complement:

1. Object NP:

   Tell [tales]

   Tell [Bob]

2. Object NP + preposition phrase:

   Tell [Bob] [about the party]

3. Object NP + finite clause:

   Tell [Bob] [that we are leaving]

It can take some other Complements too. Today, we are interested in patterns 2 and 3. Let's look at pattern (2) first.

(1) Object + Preposition Phrase

The verb TELL takes preposition phrases headed by the preposition about. The preposition phrase includes the preposition about and also a Complement of the preposition. In the example below, the complement is the noun the party

• about [the party]

Different prepositions take different types of complement - just like different verbs do. So, for example, the preposition until can take noun phrases, adverbs and finite clauses:

• Wait until [next Christmas]. noun phrase
• Wait until [later]. adverb phrase
• Wait until [she finishes her homework]. finite clause

In contrast, the preposition on can take noun phrases, and interrogative (question) clauses. It cannot take adverbs, and it cannot take normal finite clauses, only interrogative ones :

• We agreed on [a plan]. noun phrase
• We agreed on [whether to go or not]. interrogative clause
• *We agreed on [exactly]. adverb phrase (ungrammatical)
• *We agreed on [they should leave]. normal finite clause (ungrammatical)

The preposition about like the preposition on can take noun phrases, and interrogative clauses, but it cannot take normal finite clauses :

• We knew about [the plan].noun phrase
• We knew about [whether they were going to be fired]. interrogative clause
• *We know about [they were going to be fired]. normal finite clause (ungrammatical)

Preposition phrases + nouns / -ing clauses:

Almost any preposition that can take a noun as a Complement, can also take a clause using an -ing verb:

• We agreed on [a party].
• We agreed on [there being a better plan]
• I am frightened of [spiders]
• I am frightened of [falling out of trees]

(2) Object + declarative content clause

The verb TELL can take finite clauses with or without the word that. It can also take interrogative clauses. It cannot take -ing clauses:

• Tell Bob [(that) the elephants have finished the buns]. finite clause
• Tell Bob [whether the elephants have finished their buns]. interrogative
• *Tell Bob [the elephants having finished their buns]. gerund-participle (ungrammatical)

The Original Poster's Question

No one had told Smith [about [ __________ a lecture the following day]]．

A．there would be
B．there being

The Original Poster's example uses an Object NP and a preposition phrase as Complements of the verb told. The preposition is the word about. We saw that the preposition about cannot take finite clauses as Complement. This means that we cannot use (A) here. This is because:

• there would be a lecture the following day

... is a finite clause. We could use this as a sentence on its own. The preposition about CAN take noun phrases as Complement, so it can therefore also take an -ing clause as a Complement. This mean that we can use option (B):

• No one had told Smith about [ there being a lecture the following day].

Notice that this -ing clause is not finite. We cannot use it as a sentence on its own:

• *There being a lecture the following day. (ungrammatical)

Now, if the example didn't the preposition about then we use a declarative content clause. This clause would be the Complement of the verb TELL though, not a Complement of the preposition about:

• No one told Smith [there would be a lecture the following day].

If there is no about here, we cannot use an -ing clause. The verb TELL does not take -ing clauses, as we saw above:

• *No one told Smith [there being a lecture the following day]. (ungrammatical)

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_{Note for grammar junkies}

_{One of the comments suggested that we could use the word how to make another version of the sentence. This is correct. Notice that I said that about can take interrogative clauses as Complement. Well we can use the interrogative word how after the preposition about:}

_{- No one told Smith about [how there would be a lecture the following day].}

_{This modern construction doesn't really use an interrogative clause. It is a special way of allowing prepositions to take a finite clauses as Complement. The word how here is a special type of subordinator that we can use in this type of situation. The word how has no real meaning here.}

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_References

_{I originally read most of this information in the Huddleston & Pullum's Cambridge Grammar of the English Language (2002).}

Really, the only correct answer is: No one had told Smith there would be a lecture the following day. Everything else (including the "about") is just froth and bubble. Perhaps your teacher needs to immerse himself (or herself) in common English a bit more.