A solid organ is an internal organ that has firm tissue consistency; it excludes organs that are hollow (e.g., those of the gastrointestinal tract) or liquid (e.g., blood). Such organs include the kidney, liver, spleen, and pancreas.
That's the standard definition, anyway. However, it's not correct. Solid organs also include the heart and lungs, which are very definitely hollow.
Also, intestine is sometimes included as a solid organ.
Also, this leaves some ambiguity as to whether things like skin and bone marrow count as solid organs.
This is one of those terms in medicine that has drifted a bit, and is now, at best, idiomatic. A solid organ is one that is hard to transplant (blood and skin are easy, so don't count), but we can do it (so not the stomach, but sometimes the small intestines; never the brain), and it's important enough that we do it a lot. In practice, the term "solid organs" always encompasses the heart, liver, and kidney; often the lungs; and sometimes pancreas.
If you are in a specialized area, the list changes: e.g., if you're talking about laparoscopic surgery, then count spleens in as a solid organ; if you're talking about the endocrine system, the adrenal gland might make the list; and in the field of fertility, the uterus is a solid organ -- making the list only now that we are getting good at transplanting it. And when we are talking about cleaning and preserving organs, rather than transplanting them, the brain does start to be listed as a solid organ.
But also, we can do stomach transplants, but it's not a solid organ. Why? Because it is very clearly a gastrointestinal organ, the traditional antithesis of the solid organ. Really, the categories should be gastrointestinal organ vs. solid organ, and we should leave the small intestine with the gastrointestinal organs. But... stomach transplants are almost never done in isolation; they are part of a multivisceral transplant, taking the stomach, pancreas, liver, and a chunk of intestines all in one go. So the stomach will become a solid organ, as soon as we get the technology to make transplanting them individually easy and common.