"Ayyaan" (اَیّان, with a fatha on the alif and a shadda on the ya) is an Arabic word used as a conditional and adverbial particle, meaning "when" and "at what time?" (that is, it is used to ask about the time at which something occurs). Since "Ayan" (ایان) is not a meaningful name in its own right, it is not correct to name a child "Ayan".
As for "Aayaan" (آیان, with a madd on the alif), this is not an established Arabic word and no clear meaning has been found for it. Therefore, it is better to name children after the Prophets (alayhimus salam), the Sahabah Kiram (ridhwanullahi alayhim ajmaeen), the Tabieen, and the Awliyaa Allah.
Allah the Almighty knows best.
Darul Ifta Al-Ikhlas, Karachi.
المعجم الوسیط: (35/1، ط: دار الدعوۃ)
(أيان)
ظرف للزمن المستقبل نحو {أيان يبعثون} وتجيء للشرط نحو
(أيان نؤمنك تأمن غيرنا وإذا ... لم تدرك الأمن منا لم تزل حذرا)