There are several different ways that Greek words can metamorphose when being transferred into the Roman alphabet. If you directly transliterate the word αἷμα (not αἵμα), "blood", you get haima.

But some English-speakers still use the practice the Romans used in antiquity, and actually adopt the word into Latin as an intermediate step before adopting it into English. The Romans would transliterate Greek αι as ae. That way, αἷμα comes out as haema.

In addition to this step, English writers often take the additional step of reducing Latin ae to English e (I'm not sure where this practice originates; it was often done in mediaeval Latin, but I don't know if that's the real reason). Other English writers don't do this. This is why you see variations like archaeology vs. archeology, encyclopaedia vs. encyclopedia, etc. I believe that in America the latter is usually the preferred option.

Anyway, if you do this to αἱμοφιλία, it becomes first haemophilia and then hemophilia. So there's your reason.

Just to fill in the step that might not be obvious out of rosemary85's excellent answer - the consonant sound "h" in (most) Ancient Greek isn't a letter. There's a diacritic (that accent mark that looks like a backwards comma) that adds it.

This is also the standard warning that, while your intentions are good, it's generally a risky endeavor to get tattoos in languages you yourself don't understand or that at least you don't have someone close to you do the translation for. Silly things happen. Regrettable things.