It is ai·thuh for the British and ee·thr for Americans.
I am watching House of the Dragon and I encountered ai·thuh instead of ee·thr being used. It is an American television series, right? The question is whether Americans don't care casually or is it the region series is based on, or something else. I observed the same thing in Lucifer, Mazikeen always pronounced it ai·thuh. Even if she was not from this planet, she was in Los Angeles. Do the cast follow their own accent or do they go with an American accent when they are working in an American Show? It may not be a big deal for Natives to pronounce it casually as they wish but it confuses us (nonnatives) a lot especially me.