One of the many popular Mantra/Stotra in Hinduism is
And as per this Wikipedia Link, it is written by Budha-Kaushika Rishi.
Ram-raksha itself, it is mentioned that Lord Shiva told बुधकौशिक about writing the Ram-Raksha:
आदिष्टवान् यथा स्वप्ने रामरक्षामिमां हर: । तथा लिखितवान् प्रात: प्रबुद्धो बुधकौशिक:
Meaning: This protective hymn of Rama, as told by Shiva (Hara) in the dream । Was written down (as told) by the enlightened (by Shiva's visitation) Budhakoushika in the morning (upon waking up)
But beyond this, I could not found much info about this sage.
Is there any scripture describe about the Life of Budha-Kaushika Rishi and his relation/devotion to Lord Rama or Ramayana?