OK, I'm not a mental health professional, so take this with a grain of salt, but...

Be there. That's really the most important thing you can do. Just be there. Be a friend. Get them to come over and watch some anime, invite them along to board game night. Just be a friend. Companionship doesn't cure bad mental health, but loneliness often makes it much worse.

Be encouraging. The problem with "cheer up" is not only that it's trite, but also that it assumes they're doing something wrong. Don't give advice, give compliments.

If someone trusts you, be willing to push back a little when they're engaging in maladaptive thought patterns. Gently, of course, but be willing to call them out, to say "Hey, you're doing that thing where you assume the worst outcome is the most likely" or whatever. But be careful with this; you're not their therapist.

Help with the little things. Are they hydrated? Did they forget to eat? Have they not slept? Did they take their meds? You can't solve mental health with good food and exercise, but conversely being tired or hungry or thirsty can amplify symptoms a lot. Encourage them to be your gym buddy or go running together, or play some paintball. If you're going out for groceries, ask if they want to come with; maybe you've got a car and they don't, and it would really help just to have someone drive them to the store and back while you're on the way.

Be careful about gift giving. Buying someone lunch is just what friends do, but don't take it too far. Often it's better to let them do something nice for you. If someone feels worthless, it can be a big deal just buying lunch for a friend. Feeling like they've tangibly improved someone else's life. Even if you're flush and they're broke, don't turn them down if they want to do something nice for you. Instead maybe use it as an excuse to bond more; "OK, you got this one, but next time we get coffee it's on me." Now there has to be a next time you get coffee.

When they want to talk about their struggles, listen. But don't try to force them to open up. Again, you're not their therapist.

Watch out for signs of struggling. If someone doesn't do well in crowds, but wants to come to a big gathering anyway, be the person who's ready to say "Hey, do you want to get out of here?" when you can see that they're at their limit. And when you bail on the party together, maybe say "Sorry guys I've got an early start tomorrow so we're heading out." A lot of the time people don't want to single themselves out, but if you make an excuse for both of you that puts the "blame" on yourself, it's less stress for them.

Don't force socialising to always be an active thing; parallel play is really important for a lot of people. It can really mean a lot just to sit in a room with them playing on your Steamdeck while they write notes for their fanfic with their music blasting.

Be patient. When someone snaps at you, or blows off plans, or says something that was maybe a little hurtful, remember that it's not really about you. They're struggling with a disease, and a lot of the time that disease is going to make them act in ways that are hurtful to other people. That doesn't mean you shouldn't call people out for their bad behaviour, but be gentle and ready to forgive. You'd want the same from them.

And above all else, remember that they're a person, not a list of symptoms, and there is no right answer that fits every person. Depending on who they are, everything I've written could be terrible advice. You have to just take your time and get to know them. When you screw up, apologize and ask how you can do better. Remember, you experience their disease in small moments; they live with it all the time.