The correct option is A has been
In order to fill in the first blank with the most appropriate linking verb, let's first understand what linking verbs are. Linking verbs are the words that link the subject to a noun or it's adjective. For example: I am a girl. Here, 'I' is linked to the word 'girl' by the word 'is', hence 'is' is the linking verb here.
In the first half of the first sentence, 'lately' means 'recently', or 'since a few days'. Hence, we must use the present perfect tense.
Option A - 'Been' is the past participle of 'be', which means 'to exist'. 'Been' is always associated with a helping verb (has/have/had) to form a proper tense. 'The weather been sunny lately' is grammatically incorrect. Though 'been' is a linking verb, it doesn't link the subject with the complement. Hence, option A is incorrect.
Option B - 'Was' is used to show that the action or the state of the subject in past tense. Here, we have to use present perfect tense and hence we can't use 'was'. Though 'was' is a linking verb, it doesn't link the given subject with its complement. Thus option B is incorrect.
Option C - 'Has' is used to show possession or it is used to show the state of being or an action in present tense. Using only 'has' explains possession of the subject. Here 'The weather has sunny' is grammatically incorrect, as 'the weather' can't possess or own 'sunny'. We need a helping verb with 'has' to form a linking verb in present perfect tense. Here, 'has' doesn't link the given subject with its complement. Hence, option C is incorrect.
Option D - 'Has been' is used to express an action or state of being of a subject in present perfect tense. 'The weather has been sunny lately' is grammatically correct. Also, 'has been' is a linking verb, that links the subject 'the weather' with its complement 'sunny'. Hence, option D is the correct answer.