Select the most appropriate option to fill in the blank.

So dangerous did the weather conditions ______ that all mountain roads were closed.

become 

The correct option to fill in the blank is became.

The sentence is in the past tense, and the verb "become" is in the simple past tense. The simple past tense is used to describe actions that happened in the past and are now finished. In this sentence, the weather conditions are described as having become dangerous in the past, and the result of this is that all mountain roads were closed.

The other options are not grammatically correct in this context. The verb "have become" is in the present perfect tense, which is used to describe actions that happened in the past and have a continuing effect in the present. The verb "becomes" is in the present tense, which is used to describe actions that happen regularly or repeatedly. Neither of these tenses is appropriate for this sentence, which describes a specific event that happened in the past.