The correct option is A Lulu, who is my sister, plans on opening her own bakery some day.
A noun clause is a dependent clause that functions as a noun. It usually
begins with what, whatever, whom, that and other such relative
pronouns. A noun clause can act as a subject, object, predicate
nominative or object of a preposition, in a sentence.
When a noun or its equivalent is in apposition to another noun/its
equivalent, its sole purpose is to provide extra information about that
noun/equivalent.
Option A is correct because "who is my sister" is a noun clause; "who" is a relative pronoun that functions as the subject of the clause and "is" is the verb. It is in apposition to "Lulu", the subject.
Option B is incorrect because "being my sister" is not a clause but a phrase; it has no subject or verb.
Similarly, option C is incorrect. Also, the phrase "being my sister" is in apposition; hence it must be placed immediately after the noun it qualifies, 'Lulu'. It must not be placed at the end of the sentence.
Option D is incorrect because it has changed in meaning from the original sentence.