Some say "he" and some say "Christ." What is the correct Greek word or phrase here?
The Greek of this text has hOUTOS DE MIAN hUPER hAMARTIWN PROSENEGKAS QUSIAN EIS TO DIHNEKES EKAQISEN EN DEXIA TOU QEOU.
As you can see from my transliteration of the text, what some translations render "priest" or "this man" is literally "this one." That is, hOUTOS is the so-called near demonstrative in Greek and thus points to something or someone who is either near spatially, grammatically or that is near in the mind of the writer. Note the use of hOUTOS in 1 John 5:20: hOUTOS ESTIN hO ALHQINOS QEOS KAI ZWH AIWNIOS. The American Standard Version renders this passage: "This is the true God and eternal life." The writer's use of hOUTOS in Heb. 10:12 is similar.
Personally, I do not see anything particularly offensive about the rendition "this priest." It seems that certain Bibles have elected to translate Heb. 10:12 this way because of the context of the aforesaid passage. Literally, however, one might say "this one" or as the ASV puts matters: "but he, when he had offered one sacrifice for sins for ever . . ."
Lastly, I might just add that English classes I've taken in the past have encouraged me to associate substantives with demonstrative pronouns. So instead of saying, "This really makes me nervous," it is more preferable to write, "This side of town really makes me nervous." In that way, your deictic symbol (i.e., finger-pointing word) is "pointing" to a person, place, or time. At any rate, there seems to be nothing wrong with translating Heb. 10:12 as "this man" or "this priest."
NWT 2013 handles the verse thus: "But this man offered one sacrifice for sins for all time and sat down at the right hand of God,"