Why are the sentences “the weather” and “weather” used before nouns as an example to refer to a specific subject or concept, rather than a general abstract concept, such as “the weather”

When we say “The xxx” in a sentence, then, you’re referring to a specific “xxx”, so the “xxx” should already have been introduced.

If the “xxx” (the hope, in this case) hasn’t already been introduced, then you could do it in the phrase, for example “She was holding on to the hope that he would ask her to marry him.

Secondly, holding something intangible like hope is a metaphorical use, and the statement doesn’t really convey much unless you embellish or qualify it. Is “keep hope alive” a better way to say what you want to say and is a well-used – and therefore well understood – expression.