Pat’s Polemics

Look me in the eye. How explain the “me”? A frozen expression from a time when English used a dative case “me”.
Now I lay me down to sleep. How explain the “me”? A frozen expression from a time when English used a reflexive “me”.
In other words, the word “me” has changed over time in its grammatical function but these expressions remain to remind of us earlier uses. Nowadays, we would say “lay myself down”, though that itself is not idiomatic since Standard English would use “lie down”, though, again through changes in the language, most people would say “Now I lay down to sleep”, further obscuring the origins of the expression.
In the case of “look me in the eye”, a cumbersome but modern way of saying that would be “look at me in my eye”.
In Spanish, the word “me” (pronounced ‘may’) is used the way it was in Old English, in what one writer calls “an involved entity”, a high-flown phrase that simply means the “me” is somehow involved in the action, as a participant who is affected by the action, is a recipient of something acted on by the verb, e.g. an indirect object. In “James gave me the ball” we can rephrase it as “James gave the ball to me”, using the modern method which is putting a preposition with the pronoun; the first way uses “me” in the old way, as a dative case. In most cases, though, English has to use a preposition like “to”, or “with”, or “on”, or “at”, as in “gave it to me”, “looked at me”, “blamed it on me”, etc.