No repair you make to pinholes in a stainless steel sink will make them disappear, which is one reason why you might want to consider replacing the sink. Another reason is that the corrosion that caused the holes may still be occurring, and the holes could widen. If you're intent on saving the sink, you might have some success by patching them with two-part epoxy, according to multiple posts on Contractor Talk.
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How Sink Holes Appear
Far from being actually "stainless," stainless steel is vulnerable to a number of chemical stains, and the most severe are those caused by chlorine and chlorides. When these chemicals contact the surface layer of stainless steel, and there's sufficient oxygen present, a complex reaction occurs that corrodes the surface layer and causes pinholes.
Salty water is one of the most common sources of chlorides in a sink. If you're in the habit of pouring cooking water down the drain and failing to adequately rinse, the standing water can easily cause pinholes. Cleaners that contain chlorine bleach are even more aggressive, and they'll cause the same type of pinhole corrosion if you use them without rinsing, and they'll do it faster.
Stainless Steel Epoxy Repair
You don't have many practical options for a hole-in-the-sink repair. Welding the hole is possible (but not practical) while patching the hole with epoxy is both possible and practical, but there's still a problem. The repair won't make the holes disappear, so you're stuck with the unsightly corrosion for as long as you keep the sink.
The repair product often mentioned by contractors considering how to repair pinhole damage is J-B Weld. This product is an opaque two-part epoxy putty that's usually gray or white, and it comes in water-resistant formulations. If you don't like the idea of an opaque putty, you might consider using a clear epoxy adhesive, such as Loctite or Gorilla Glue two-part epoxy.
If the pitting hasn't penetrated the metal, you should use a water-resistant product because you'll have to apply it to the side of the sink that holds water. If the pinholes have penetrated, that can actually be a good thing. In this case you can do the patching from under the sink where the epoxy won't be visible.
How to Apply the Epoxy
Epoxy adhesive is a liquid, although a highly viscous one, and you easily can force it into the pinhole depressions with a putty knife immediately after mixing it. You can also do this with epoxy putty, although it's stiffer and harder to spread. Immediately after filling the holes, clean excess glue or putty with a rag and some acetone to make the repair look as good as it possibly can.
If the holes have penetrated, it's easy to find them by going under the sink and looking for daylight on the underside of the sink. Apply epoxy putty to each of the holes you find by spreading it liberally on the underside of the sink with a putty knife. Wait for the cure time specified on the epoxy label before testing the sink.