What is the meaning of "argue about words" in 2 Timothy 2:14?
Paul is speaking about the apostasy that would occur. A few verses later, Paul even mentions two individuals, Hymenaeus and Philetus, that were deviating people from the truth about the resurrection.
Note what the Watchtower July 15, 2014 article entitled "Jehovah's People 'Renounce Unrighteousness'" mentions:
God’s Word gives specific advice concerning the types of unrighteousness that Christians must renounce or reject. For instance, in the immediate context of 2 Timothy 2:19, we find that Paul tells Timothy “not to fight about words” and to “reject empty speeches.” (Read 2 Timothy 2:14, 16, 23.) Some members of the congregation were promoting apostate teachings. Also, it appears that others were introducing controversial ideas. Even if the latter were not directly unscriptural, they were divisive. They resulted in bickering and arguing over words, creating a spiritually unhealthy atmosphere. Hence, Paul stressed the need to “reject foolish and ignorant debates.”
The Watchtower September 1, 1980 article entitled "Overseers–Be Fine Examples to 'the Flock'" also touches on this:
Paul told Timothy to charge others “not to fight about words, a thing of no usefulness at all because it overturns those listening.” The apostle then referred to the unhealthy effects of statements made by men who had deviated from the truth. There is nothing to be gained and likely much to be lost spiritually by fighting about words or getting involved in debates with those who reject healthful teaching. (2 Tim. 2:14-19; Titus 1:7-9) This does not mean that a person cannot sincerely raise a question about a matter that he does not understand. However, insisting on some viewpoint may cause unnecessary disturbance. Of course, if some individuals need spiritual help, there is a proper way to teach, and that is “with mildness.”—2 Tim. 2:23-26.
[All scripture quotations from the New World Translation of the Holy Scriptures (Study Edition)]