The chief argument against Ephesians 2:8-10 is James 2:24. Many people should already have picked up on how I phrased that. I quite deliberately stated, rather than "the chief argument against salvation by faith alone", that it is the chief argument against Ephesians 2:8-10. This is because Ephesians 2:8-10 is among the clearest passages supporting salvation by grace, through faith, not by works, as a gift of God. In fact, it explicitly says "For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast. For we are his workmanship, created in Christ Jesus unto good works, which God hath before ordained that we should walk in them."
How much clearer could God have made it in just these three little verses? Salvation is a gift from God, offered by His grace, received through faith, not of works. Verse 10 even goes so far as to basically say that the good works are a result, and indeed the very purpose, of salvation, not the cause thereof.
Yet, in spite of this, many heretics preach a "gospel" that includes works in salvation somehow. The go-to verse defending these fake gospels is James 2:24, which is the only verse in the entire Bible that says the words "faith alone", or "faith only". Thus, heretics claim, it disproves Sola Fide (salvation by faith alone), because it says we are justified by works, and not faith alone.
But what's that word? Justified. In Ephesians, the word is saved. We are saved by grace, through faith, and not of works. But are we justified by faith alone? James refutes this idea by giving examples. Abraham "believed God", and that was credited to him as righteousness (i.e. he was saved by faith alone), yet when he eventually offered his son, Isaac, that is when he was justified. Well, turn back to Genesis 22:12. Here, just before Abraham is about to sacrifice Isaac to the Lord, the Lord commands him "Lay not thine hand upon the lad, neither do thou any thing unto him: for now I know that thou fearest God, seeing thou hast not withheld thy son, thine only son from me." (emphasis mine).
This is what justification means. You could say it like this: justification is adding bite to your bark. Anyone can say they have faith, but if they truly have faith, they will show that faith. Justification is all about proving something. This is why God Himself can be justified by sinners (Luke 7:29), and even wisdom, an immaterial concept lacking consciousness, can be "justified by her children" (Matthew 11:19; Luke 7:35).
Thus, we see that James 2:24 is an erroneous argument against Sola Fide. Everything else in Scripture testifies that salvation is by grace, through faith, and that not of ourselves, it is a gift of God, not of works, lest anyone should boast. Thus, unless we believe James is some uninspired false prophet who somehow got his book included among the word of God, he is not saying "salvation is by grace, through faith and works, as a wage rather than a gift, therefore go forth and boast". Rather, James' message is "if you're saved, prove it. Act saved."