The sleep mudra is the thumb and the first (index) finger are gently joined, is known as Gyan Mudra.

This will soon be supported with a credible source to confirm the benefits of this mudra for sleep.

This mudra is used to balance the mind and help with sleep:

The Gyan Mudra (Mudra of Knowledge) or Mudra of Consciousness.
How it’s performed: Gently touch the tip of the thumb to the tip of the index finger, while the other three fingers are extended and relaxed.

Benefits related to balancing the mind and aiding sleep:

Calms the mind and reduces stress/anxiety: Numerous sources state that Gyan Mudra helps to alleviate stress, anxiety, nervous tension, and promote a sense of calm and relaxation.
Balances the nervous system: According to Ayurvedic principles, Gyan Mudra helps balance the Vata dosha (associated with air and space elements), which directly relates to the nervous system. A balanced nervous system is crucial for restful sleep.
Alleviates insomnia and improves sleep quality: Many sources explicitly mention Gyan Mudra as beneficial for those struggling with insomnia or other sleep disturbances, as it promotes relaxation and prepares the mind for restful sleep.
The consistent theme across various yoga and mudra resources is that Gyan Mudra helps to calm the mind, reduce mental chatter, and balance energy, all of which are conducive to achieving better sleep.