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.