Unless a nonprofit membership organization is so large and open as to be considered a “public accommodation” subject to anti-discrimination laws, the general rule is that it may choose its own members as it sees fit.  But even if your group is subject to public accommodation laws, blowhards who threaten frivolous litigation are not a protected class and would not be protected against rejection for that reason. 

You need to check your bylaws and your policies, however, to see how people are admitted to membership.  If you say anyone who pays $10 and shows up is automatically a member, you would probably have to change your rules to keep this person out.  In that case, you may want to consider whether you have an adequate procedure for removal. 

If you have an actual procedure for admission, like a social club, you may be able to treat renewal as a re-admission and deny re-admission on the basis of the unwelcome conduct.  In order to protect against a hostile takeover or having to deal with unwanted jerks, we provide in our form bylaws that the board may deny membership to individuals for “any reason” even though they meet the general qualifications for admission.  (See Ready Reference Page: “Bylaws Function as ‘Constitution’ of Nonprofit Corporations”)  From your description of this person as a “former member,” it sounds as though you would have the power to deny re-admission.

If you have a practice of automatic renewal and this person is just paying dues to continue, it may be more difficult to deny membership.  But if you really don’t want this person to be a member (and if you have the votes), I would provide the person with the opportunity to plead his or her case before the board before making a final decision.  Courts are very unwilling to overturn the internal decisions of membership organizations if there is some semblance of “due process” in disciplinary-type actions.