#1/#2. The correct verb to use is "is". "Is" is the 3rd-person singular present tense form of the verb "to be", while "are" is the plural present tense form of the same verb. ("To be" is widely considered to be the most irregular verb in the whole of the English language).
Since the subject of the sentence ("her only hope") is singular, the verb must also be singular in order to be in subject-verb agreement. The object of the verb "her children" does not have to be in agreement with the subject and verb.
#3/#4: Both sentences are grammatically correct, although #3 sounds more natural, at least to this native speaker. For the same reasons as #1/#2, it is grammatically incorrect to use "are" (but a native speaker may so anyway). However, we could say "Our crops are principally potatoes", and retain the same meaning.