Question about Neeson's character?

The actor is Irish. :B Guess it slips through.

You missed something. When he introduces himself to the psychologist and they're getting to know each other while observing Nell, he explains that he is the only child of Irish parents, unlike the neighbors there who were destitute and had something like seven kids.

The original script had him an Irish-American from New York who had worked extensively with terminally ill children. He'd messed up slightly on a medication order for one of them and she died a bit before she was expected to. Lovell blamed himself. He was in Robbinsville to get as far away as he could from everything he had known and went into general practice there.

Whether this was done to accommodate Neeson or whether he'd even been cast at that point I don't know. But it makes sense to have someone with an Irish background in the story, since Irish people are perceived (at least in the movies!) as being a bit more open-minded to things that appear unusual to others or which seem a bit otherworldly. The environment as well as Nell herself had this kind of appeal.

It shouldn't really matter all that much. We have people from all over the world living in all parts of the country (US and Canada), why would it be such a stretch to think that a doctor in a small town would have an Irish accent? It did not stand out as being strange to me at all.

He had an Irish accent because, near the beginning, he had mentioned that he had come from an Irish family. It may have been to accomodate him as an irish actor..who knows.