The Michelin attorneys have brought an action to cancel your registration of the domain using arbitration procedures established by ICANN, the organization that oversees domain name control. Under these procedures, Michelin can only cancel the domain if it demonstrates to an arbitrator that you acted in bad faith. The company has filed a complaint dated October 2, 2015 and you have 20 days to file a response. The rules for responding are explained here.
Did you act in bad faith? Under the arbitration rules, four ways to demonstrate bad faith include:
(i) the domain was registered primarily for the purpose of selling it to the complainant or a competitor for more than the documented out-of-pocket expenses related to the name; orWe assume Michelin is basing its claim on the fourth item. Under arbitration rules, an acceptable defense is if "you are making a legitimate noncommercial or fair use of the domain name, without intent for commercial gain to misleadingly divert consumers or to tarnish the trademark or service mark at issue." Under these standards, it does not appear as if you were acting in bad faith because your mother's blog has no commercial purpose (typically, the strongest factor); there is little chance of diversion as Michelin customers are sophisticated enough to realize that the blog is not connected with Michelin (as is obvious from reading the blog's mission); and you are making a nominative fair use of the Michelin mark.
(ii) the domain was registered in order to prevent the mark owner from using it, provided that the registrant has engaged in a pattern of such registration; or
(iii) the domain was registered primarily to disrupt the business of a competitor; or
(iv) by using the domain, the registrant has intentionally attempted to attract users for commercial gain by creating a likelihood of confusion as to source or affiliation.
What to do? Although we don't believe you are acting in bad faith, our opinion may not matter much as such arbitration rulings are difficult to predict, often biased in favor of the complaining party, and will likely require an attorney's assistance. At the same time, the lack of a response is sometimes even used to infer bad faith as was done in Michelin's complaint ("Respondent never answered the Complainant's letter is further evidence of bad faith ..."). For that reason, if you feel strongly about asserting your rights, you should fashion and file a suitable response to be filed with the arbitrator.
On the other hand, you may be able to say goodbye to the whole thing by contacting Michelin and simply agreeing to retire the domain. As you are aware, your mother's blog can exist without the domain name. To lower your chances of being hassled over the blog, a prominent disclaimer may help -- something to the effect that the blog is not endorsed by or associated with Michelin or the Michelin Star.