Mercury capsules had names (e.g. Liberty Bell).
Apollo command modules had names (e.g. Columbia).
Apollo lunar modules had names (e.g. Challenger).
Space Shuttle orbiters had names (e.g. Columbia and Challenger).
Why didn't the Gemini capsules have names?
Actually, the first manned one (Gemini 3) did have a name, "Molly Brown1".
According to an article in Life Magazine, Oct 11, 1968, "Spacecraft Anonymous", upper NASA management became weary of arguing about spacecraft names with the crew after Grissom decided to name his capsule that.
1 "Molly Brown" was a reference to the movie and play "The Unsinkable Molly Brown" which is a joking reference to the sinking of Grissom's capsule on his Mercury flight. The title character is a survivor of the sinking of the Titanic.
Supplemental to Organic Marble's answer:
The Gemini no-names policy remained in effect through the early part of the Apollo program as well; the Apollo 7 and 8 spacecraft were nameless. The names returned when Apollo missions started flying two spacecraft independently; for the CSM-only missions, the call sign was simply "Apollo", but with a CSM and LM both on the air, they needed separate call signs. The pre-landing spacecraft got somewhat irreverent names ("Gumdrop" and "Spider" for A9, "Charlie Brown" and "Snoopy" for A10), but the remaining Apollo flights were named with management-acceptable levels of gravitas.
After the landings, the Skylab and ASTP flights went back to generic call signs. I believe the Skylab call signs were "Skylab 2/3/4" even while the crew was in the CSM, but I'm not certain.
Once NASA was flying reusable spacecraft, it made more sense to give them individual names.
