You can use "what" at the beginning of a sentence for two purposes:

1. to ask a question:

What is a better way to spend the day than at the beach (= I don't want to spend my day at the beach. Is there a better way to spend my day?)

2. to emphasize your idea and to indicate that the thing you are talking about is the best.

What better way to get inspired than by accompanying a good friend? (= There is no better way to get inspired than by accompanying a good friend)

In your sentence, when he says:

But the more I played with the idea, the more I thought, What better way to ward Juli off?

he conveys that "His idea is the best and there is no better way to ward Juli off than his idea".

You can use "what" at the beginning of a sentence for two purposes:

1. to ask a question:

What is a better way to spend the day than at the beach? (= I don't want to go to the beach today. Is there a better way to spend my day?)

2. to emphasize your idea and to indicate that the thing you are talking about is the best.

What better way to get inspired than by accompanying a good friend? (= There is no better way to get inspired than by accompanying a good friend)

In your sentence, when he says:

But the more I played with the idea, the more I thought, What better way to ward Juli off?

he conveys that "His idea is the best and there is no better way to ward Juli off than his idea".

You can use "what" at the beginning of a sentence for two purposes:

1. to ask a question:

What is a better way to do this?

2. to emphasize your idea and to indicate that something you are talking about is the best.

What better way to get inspired than by accompanying a good friend? (= There is no better way to get inspired than by accompanying a good friend)

In your sentence, when he says:

But the more I played with the idea, the more I thought, What better way to ward Juli off?

he conveys that "His idea is the best and there is no better way to ward Juli off than his idea".

You can use "what" at the beginning of a sentence for two purposes:

1. to ask a question:

What is a better way to spend the day than at the beach (= I don't want to spend my day at the beach. Is there a better way to spend my day?)

2. to emphasize your idea and to indicate that the thing you are talking about is the best.

What better way to get inspired than by accompanying a good friend? (= There is no better way to get inspired than by accompanying a good friend)

In your sentence, when he says:

But the more I played with the idea, the more I thought, What better way to ward Juli off?

he conveys that "His idea is the best and there is no better way to ward Juli off than his idea".