al-Kāfi: Moḥammed bin Yaʿqūb, from a number of our companions, from Aḥmed bin Moḥammed bin Khālid, from al-Qāsim bin Yaḥya, from his grandfather al-Ḥassan bin Rāshid, from ʿAbd Allāh bin Sinān, who said:

I asked Abī ʿAbd Allāh عليه السلام about the exegesis of In the name of Allāh, The Entirely Merciful, The Especially Merciful (Bismillah al-Raḥmān al-Raḥīm), where he replied: The letter bā’ (الباء) [in bism (بسم), stands for the] Bahā’1 of Allāh.
The sīn (السين) [stands for] Sanā’2 of Allāh.
The mīm (الميم) [stands for] Majd3 of Allāh.
Some have narrated that the mīm (الميم) [stands for] Mulk4 of Allāh.
Allāh [means that he] is The God of all things.
The meaning of The Entirely Merciful (al-Raḥmān), [is that] His Mercy embraces all creation.
The meaning of The Especially Merciful (al-Raḥīm), [is that] His Mercy is exclusively for the Believers.5

1. Beauty and Magnificence.
2. Splendour and Radiance of Beauty.
3. Glory and Honour.
4. Sovereignty and Authority over all creation.
5. Sayyid ʿAbd Allāh al-Sabzawāri رحمه الله has stated in his tafsīr Muwāhib ul-Raḥmān, Volume 1, pg. 27, that the knowledge of letters is an honourable science, whose knowledge rests only with those invested with authority from Allāh, the Exalted, and are aware of the inner realities of things – and they are the Purified Household عليه السلام.

محمد بن يعقوب : عن عدة من أصحابنا ، عن أحمد بن محمد بن خالد ، عن القاسم بن يحيى ، عن جده الحسن بن راشد ، عن عبدالله بن سنان ، قال : سألت أبا عبدالله ( عليه السلام ) عن تفسير * ( بِسْمِ اللَّه الرَّحْمنِ الرَّحِيمِ ) * ، قال : الباء بهاء الله ، والسين سناء الله ، والميم مجد الله – وروى بعضهم : الميم ملك الله – والله إله كل شيء ، الرحمن بجميع خلقه ، والرحيم بالمؤمنين خاصة