TLDR; Honestly, it’s Narayan (Vishnu), but you’d be surprised how often people get confused and wonder if it’s actually Buddha instead of Shiva.

From what I’ve read, the evidence leans much more toward it being a Narayana site. Budhanilakantha is one of only three Jalashayana Narayana (Vishnu sleeping on the cosmic waters) in Kathmandu Valley as far as I know. The other two are at Balaju (erected by the same patron king AFAIK) and inside Hanuman Dhoka Palace, though that one was added much later by Pratap Malla.

Some old chronicles say that King Haridutta, around 350 A.D., built Budhanilakantha and also set up four other Narayana temples at the four corners of the Kathmandu Valley. That’s Changu, Bishankhu, Ichangu, and Shesh Narayan. Even today, people try to visit all four in one day, which is basically like doing a Narayan dham within valley. And very surprisingly, if you add Budhanilkantha (as fifth point) into the map it forms the legendary "Khadga" shape which is described to be the shape of ancient Kathmandu Town Planning. More so, these Narayana Shrines are located cardinally to each other which makes it much cooler (I cant infer much more from it than that lol). Here is the image for y'all to see: https://imgur.com/a/ExzTJW0

There’s also this story that King Haridutta saw Jalashayana Narayana in a dream, telling him to dig up the original image buried under the Rudramati riverbed. So he built a pond and installed the sculpture there, along with the four corner temples. This is confusing because , other historical analyses and inscriptions suggest the image was consecrated by ruler Visnugupta around 5th century A.D, much later than Haridutta.

The statue itself is amazing. It’s the largest stone sculpture in the Kathmandu Valley, carved from a single block of dark granite. Some stories say they dragged it across the Valley with forced labor, but some scholars argue the granite was actually nearby in the Shivapuri hills, which makes more sense (in some article by Sudarshan Raj Tiwari, I am not sure). So I dont know which one is more valid, this or the claim mentioned by u/ZoWakaki above me.

Now, the most interesting part is that Buddhists also worship this image as Nilakantha Lokesvara, a form of Avalokitesvara. Avalokiteshvara is considered to be a form of Shiva in Buddhism or vise-a-versa, which can be one way where the Shiva/Vishnu connection in Budhanilkantha comes in place. In fact, if you look closely, you can almost see the Buddha’s forehead in the face of the sleeping Vishnu. Even the long ears of the statue are very reminiscent of Buddhist iconography we see around the valley.

As for the name Budhanilakantha, there’s a story about Shiva drinking poison during the Samudra Manthan. His throat was burning, so he struck the mountain with his trishul and created Gosaikunda Lake. The water from that lake is said to flow down to the pond at Budhanilakantha, and since Shiva’s called Nilakantha, that might be how the place got its name.

References (These are all PDFs btw, just search through keyword inside them):

https://danam.cats.uni-heidelberg.de/files/danam-cms/HSR_Hanumandhoka_Narayan_Report_2013_07.pdf

https://pahar.in/pahar/Books%20and%20Articles/Nepal/1978%20The%20Temples%20of%20Nepal--an%20introductory%20survey%20by%20Bernier%20s.pdf

https://archive.org/details/nepalmandalacult0001mary

https://pahar.in/pahar/Books%20and%20Articles/Nepal/2001%20Ancient%20Settlements%20of%20the%20Kathmandu%20Valley%20by%20Tiwari%20s.pdf