Changing your username and avatar are not the only options. People can also change the text in the 'about me' section of their user page. Note that if your reputation is >1k, you can have an expandable usercard that will show some of the text from your about me section when someone mouses over your username on a post.
Some suggested text was first proffered here:
The recent actions of Stack Exchange have been reprehensible. The lack of professional conduct, the tone deaf way they've been dealing with the community, and the unfair and unjustified way they've dealt with Monica Cellio.
Monica is an exemplar of the community, it's incredible how she's managed to keep a level head during this time when emotions have been very high.
Update: the gofundme campaign has been closed.
A newer option has been proffered here. What has changed is that Monica has started a gofundme campaign to offset the costs of legal action. As a result, people might change their username to append "go fund Monica", instead of "reinstate Monica". The suggested about me text is then:
Consider supporting Monica Cellio's gofundme here >>>
A popular and well-respected volunteer network moderator, Monica Cellio, was terminated without warning and portrayed to the press as misgendering trans people by Stack Overflow, Inc. All signs suggest this was an erroneous, if not malicious, decision not founded on fact, which the company is completely failing to take any responsibility for. This has caused an enormous and ongoing uproar in the community.
Monica and the community have been quite literally begging for some kind of response from the company either opening up a path to reinstatement, or justifying their decision to terminate her, for more than a month now. She's made many good-faith offers to talk, and others have offered to mediate (and made dozens of constructive suggestions on how to resolve the situation). Members of the Lavender community have spoken up on her behalf (1, 2).
So far, Stack Overflow, Inc. has met all of this with thunderous silence. As a long-time contributor and someone always willing to see the good in the company's and its employees' actions and assume good faith, I'm still struggling to come to terms with the corporate callousness of this behaviour which we have to assume now is the new normal.
Having exhausted all other means of communication, Monica is now raising funds for potential legal action to clear her name. This is in her own interest, but arguably also in the interest of the SO community at large. While only the libel is likely to be legally actionable, it might establish at least some boundaries to what the company can do to its volunteers, as well as be a symbolic slap in the face of a business that appears to have lost all respect for the community that keeps it alive.
I have complete trust the funds will be used well and anything left over (say because they decide to do the right thing after all and legal action becomes unnecessary) will go toward good causes.
Consider supporting her today.