At what stage of life is oogenesis initiated in a human female?

The correct answer is Option (4) → During embryonic development.

Oogenesis is initiated before a girl is born. During embryonic development a couple of million oogonia are formed in each fetal ovary, and no more oogonia are added after birth.

Option (1) During the menstrual cycle. — Incorrect. The menstrual cycle starts only after puberty, long after oogenesis has begun.

Option (2) At puberty. — Incorrect. Many primary follicles degenerate between birth and puberty, so the follicles, and the oocytes in them, are already present before puberty. Puberty only marks the start of the menstrual cycles.

Option (3) During menarche. — Incorrect. Menarche is the first menstruation, which begins at puberty. It is not the time when oogenesis is initiated.

Option (4) During embryonic development. — Correct. Oogonia form within each fetal ovary, start dividing and enter prophase-I of meiosis, where they stay arrested as primary oocytes.

NCERT Extract:
Gametogenesis – “… The process of formation of a mature female gamete is called oogenesis which is markedly different from spermatogenesis. Oogenesis is initiated during the embryonic development stage when a couple of million gamete mother cells (oogonia) are formed within each fetal ovary; no more oogonia are formed and added after birth. These cells start division and enter into prophase-I of the meiotic division and get temporarily arrested at that stage, called primary oocytes.”
Menstrual Cycle – “The reproductive cycle in the female primates (e.g. monkeys, apes and human beings) is called menstrual cycle. The first menstruation begins at puberty and is called menarche.”

Figure 2.8 (b) Schematic representation of Oogenesis