What type of grammar construction is this

It seems to me that all three have the same meaning. I would criticize all of them for wordiness. In my opinion, the third sentence is the worst whereas the second one is the best. However, I would eliminate "is able to" from the second in favor of can. Something like: "Maria believes she can best him in this argument," would be my favorite.

The first sentence conveys the meaning that she thinks/knows she can best him in the argument. Whereas the last sentence is little bit skewed in the meaning. The sentence conveys that she has to think to best him in the argument. The last sentence rhymes with meaning of the sentence "she prepares herself to be able to best him in this argument.

Each of these sentences has two proposition:

There is an argument
She thinks she can best him in it

All three of them are functionally the same. Another would be 4. she thinks she is able to best him in the argument

1. Uses the object form (which must be the -self form as it refers to the subject) and elides "to be"
2. Uses the subject form
3. Uses the object form with "to be"
4. Uses the subject form without "that"

2 and 4 are just like "I said (that) he stinks" which works with or without "that".

Anyway, all four of these sentences describe an identical function and the truth of each one describes an identical universe, so ya, all three are the same in meaning.

As Kris notes, the first sentence differs from the third only in the presence or absence of to be:

• She thinks herself (to be) able to do it.

This is an example of the syntactic rule To Be-Deletion, which applies to Infinitive Complements; and the main verb think can take an infinitive complement clause, so it's grammatical.

To Be-Deletion applies to infinitives with a predicate adjective (here, to be able ...)
and deletes the predictable complementizer to and the predictable auxiliary infinitive be,
leaving only the predicate adjective phrase (able ...) following its subject (herself).

So (1) = (3).

(2), however, while it means the same thing as the others, is a different construction. The main verb think can also take a tensed That-Complement. This is just a declarative sentence with the complementizer that marking it as subordinate.

• She is able to do it. (main clause, complete sentence)
• She thinks (that) she is able to do it. (subordinate clause)

And, as McGurk points out, the that is deletable, because it's predictable,
much like to be, but operating by a different rule.