The correct sentence should be:
"What did he say his name was?"
Your doubt is pretty reasonable as his name is, was and always will be the same.
Then why is it correct to use 'was'?
Because this is an example of reported speech.
You're involving a third person in the conversation. You're telling him about a conversation you had with another individual, in other words, you're reporting to him.
The thing with reported speech is that the tenses always change.
We don't need to change the tense at all times, though probably we do need to change the 'person' from 'I' to 'she', for example. We also may need to change words like 'my' and 'your', but in general, the tense in reported speech is one tense back in time from the tense in direct speech.
Even though the subject of discussion exists or is valid in the time of reference, we use past tense in reported speech.
He told me that he liked chocolate.
Even though 'he' still likes chocolate, the original conversation happened in the past. So this is a reported speech.
This link tells you more about why and how the tenses change in reported speech.