Word meaning “convalescing” their mental health

Solution 1:

Revive might fit in some cases, though it has other connotations as well.

to bring or come back to mind

cause to regain consciousness; "The doctors revived the comatose man"

Note: There are psychiatric comas and structural/metabolic comas.

Perk (or perk up) can be used in the case of depression:

to gain in vigor or cheerfulness especially after a period of weakness or depression

—usually used with up: he's perked up noticeably

It is used in psychology related sources as well. It does not usually mean full recovery, it usually means feeling better in the period of mental illness. There are details and colloquial usages also.

Note: We were trying to find a verb other than "recover" and OP asked for alternative verbs that is only related to mental health. I already mentioned that the answers that I have can be used in certain cases but talking in details lead to a medical discussion.

Solution 2:

I am not a medical expert, but in standard American English there's a word called recover, which means to get back to normal after a sickness.

Solution 3:

I feel that it's important to put some perspective here on language used with mental illness.

Perk up is not a synonym for recovery. There is no language of recovery unique to mental illness. It just does not exist. If you want my background, hover over the empty field.

_{Background: For a short time, I ran a free mental health and addiction clinic. I was certified through the Substance Abuse and Mental Health Services Administration (which is the Federal Government's lead agency for improving the quality and availability of substance abuse prevention, addiction treatment, and mental health services in the United States.) Psychiatrists and psychologists already have the training to capably run such clinics, but I am neither. Therefore, SAMHSA provides training for other kinds of professionals, and certification on completion of many hours of specific coursework and passing the exams.}

I can tell you that the goal was recovery, not perking up. Reintegration. Perceived helpfulness. Effective treatment. Improvement in levels of functioning, decreased levels of cognitive and functional impairment.

There is no word here specific to mental illness. When it is treated as an illness, it will have terminology similar to that of other illnesses.

Your wish to the contrary doesn't make a trivial expression sufficient to represent recovery from a broad category of mental illnesses.

_{Also, we are not as pessimistic as Fumble Fingers. People do recover from Depression, PTSD and others. People can be functional and integrated with bipolar illness and schizophrenia. We do not use remission (we leave that to oncologists). We do use stable, We believe that mental health is like physical health. SAMHSA has a Recovery Month (spreading the positive message that behavioral health is essential to overall health, that prevention works, treatment is effective and people can and do recover) but no remission month. And please, when people talk about slapping their doctors, it makes me a bit anxious about going to work.}