I would say "hands" without a doubt.
The "standard" rendition of the first half is, very literally:
καὶ δοθήσεται τὸ βιβλίον τοῦτο
And this book will be given
εἰς χεῖρας ἀνθρώπου
into [the] hands of a person
μὴ ἐπισταμένου γράμματα…
[who is] not knowing letters…
Or, using more idiomatic English, "and this book will be placed in the hands of one who cannot read". (It's entirely reasonable Greek, it just sounds awkward when I try to keep every Greek word intact in English.)
Given the context, ειϲ χιραϲ ανθρωπου has to mean "into [the] chiras of a person". Using a proper noun here, "into the Hira of a person", makes less sense than assuming a simple mistake—swapping ι for ει is an extremely common error from the Koinë period onward, since they were pronounced identically.
EDIT: Google has led me to a very…eccentric…theory that this passage is meant to say "will be given to a man at Mount Hira". I'm guessing you've found that too, and that's where this question is coming from.
But I'm quite confident that's not the case here:
- Εἰς very clearly means "into", indicating a direction of motion; I've never seen a situation where it can mean "at" or "in".
- Proper names, especially foreign ones, almost always take a definite article—which is lacking here.
- Ἀνθρώπου is very clearly in the genitive, and εἰς always requires an accusative.
- The Greek letter χ isn't generally used for an H-like sound in the Septuagint: look at 2 Samuel 23:26, where the name "Hira" is written Εἴρας. Using χ where it doesn't belong could be scribal error, but if so it's not anywhere near as common as ει → ι.