I'm sure there is a way, but so help me, I can't find it.

I can't seem to give my companion armor or weapons or potions. My Lydia was complaining that she was hurt. So I tried to give her a couple health pots. No dice. There was no way to do that via the dialogue boxes as far as I can tell. Then I tried dropping a health pot on the ground. Nope, she wouldn't touch it. So then I tried sneaking and reverse pickpocketing a health pot into her inventory. Nope. This is getting frustrating.
How can I give my companion weapons or armor or potions?

Talk to her. There's an option for "I need to trade some items with you" or something like that.

By the way, if you're not in combat, her health will regenerate on its own.

Tell her you want to trade stuff, or something similar, dont remember what it is called. You can then do the same thing that you do when buying/selling with a merchant, except no money involved. Ie, click something in your inventory and she will get it, click something in her inventory and you will get it.

There are a ton of way to interact with your companions. As pointed out above, you talk to them and say, "I'd like to trade with you." and you can swap items from your inventory to theirs or vice versa.

Also you can say, "I need you to do something for me." and you can direct them to do almost ANYTHING in the environment, from attacking someone to stealing a specific item to sitting in a chair to walking through a door, etc. etc. etc.

I played with this for quite some time. The only thing I could do through that menu was say "wait here". But I'll look at it again.