“You should wake me,” I say, thinking about how I can interrupt his sleep two or three times on a bad night.
You should wake me,” I say, thinking about [how I can interrupt his sleep two or three times on a bad night].
"About" is best analysed as a preposition, and "how" as an interrogative adverb.
The expression "how I can interrupt his sleep two or three times on a bad night" is a subordinate interrogative clause (embedded question) functioning as complement of "about".
The meaning is "... thinking about the answer to the question 'How can I interrupt his sleep two or three times on a bad night?'"
... I say, thinking about how I can interrupt his sleep two or three times on a bad night.
Everything following the preposition 'about' is part of a prepositional phrase.
Prepositional phrases require an object such as a noun, noun phrase, or pronoun; or a complement.
I can substitute the pronoun 'it' into the sentence in place of the entire clause, in other words...
"I say, thinking about it..."
"how I can interrupt his sleep two or three times on a bad night"
is the complement of about, and an embedded question. AKA subordinate interrogative clause.
The original question would have been:
How can I interrupt his sleep?
How is an interrogative adverb seen here in an inverted SV structure , but the entire clause from the OP is in its non-inverted form as the complement of the preposition 'about'; this is normal with embedded questions.
The non-inverted construction is used because it is not a direct question.
"About" is a preposition; the reason for that is that what follows "about" is a nominal clause (or noun clause, alternatively put), and that such a clause, which is similar to a noun, can stand as the prepositional complement (CoGEL p.655 § 9.1, "The prepositional complement is characteristically a noun phrase, a nominal wh clause, or a nominal -ing clause."); "How" is one of the so called wh words.
- nominal clause after "about": how I can interrupt his sleep two or three times on a bad night
In this clause "how" is an adverb, it modifies the verb 'interrupt". It is also considered to be a relative word; this is so because all nominal relative clauses "can be paraphrased by a noun phrase containing a noun head with general reference that is modified by a relative clause" (CoGEL p.1056 § 15.8). Such an equivalent could be as follows.
- the way in which I can interrupt his sleep two or three times on a bad night
The word "way" is the general reference. "In which …night" is the modifying relative clause.
From CoGEL p. 1057 § 15.8, "the wh element may function within the relative nominal clause as subject, direct object, complement, adverbial, or prepositional complement.
Since "how" is an adverb modifying "interrupt" (see for instance Noun-Clause Starters, Thought Co, how is an adverb modifying the verb), it functions as an adverbial modifying the verb "interrupt".
In view of the criticism that this answer received, given the unconditional rejection that can be read into it and given its being hermetic to all reasoning, I will add some references here to show that besides those I have mentioned above there are others that share the same point of view, a traditional one in English grammar.
The clause can't be but subordinate to "itself", since it contains its antecedent. It is what is also called a free relative clauses: "in English grammar, a free relative clause is a type of relative clause (that is, a word group beginning with a wh-word) that contains the antecedent within itself" No antecedent in the main clause, no subordination to that clause) (free relative clause, Wikipedia)
English allows what is called a free, fused or nominal relative construction. This kind of relative construction consists of a relative clause that instead of attaching to an external antecedent—and modifying it as an external noun phrase—is "fused" with it; and thus a nominal function is "fused" into the resultant 'construction'. For example:
- What he did was clearly impossible.
Here "What he did" has the same sense as "that which he did", or "the thing that he did". Thus the noun phrase the thing and the relative pronoun that are 'fused' into what; and the resulting relative construction "What he did" functions as the subject of the verb was. Free relative constructions are inherently restrictive.
References suggested and presented by user Pablo GM (6/5/21)
For further reading, see fused relatives versus interrogatives. You can also check this other post on relative pronouns after a preposition . This other post on fused structures can also help illustrate the issue. In it, there is an answer with a recommendation to read Dixon's Basic Linguistic Theory, sec. 17.5.3, which is a good read.