Roth, no. Traditional IRA, maybe

Did you use google:

are foreign earnings eligible for ira

Because there is quite the results list for that issue, and you can read all about it.

Roth does not reduce taxable income. Whether they are eligible for Roth IRA or not, is a different question. Traditional IRA is how you make deductible contributions. Whether they are eligible for Trad IRA or not, is a different question.

What factors determine eligibility for this situation?

You can't contribute based on excluded foreign earned income.

If he makes above the foreign earned income exclusion limit, he can contribute based on the "excess" earned income.

"What factors determine eligibility for this situation?"

The search results you will get include IRS guidance for how foreign income is computed for purposes of retirement account eligibility. Did you do that search and read some of those topics?

google: are foreign earnings eligible for ira

Results include IRS links. Did you want that copied and pasted here, or perhaps it is more useful to read them yourself, and perhaps bookmark those that are good reference for you to use later.