She sort of gave you half the answer already. And you already have half the intention. So you are in a good track.
With all those conversations, even when sometimes leading to fights, you should already know a lot about what she likes and what she would consider a loving or caring gesture.
I don't think she's asking for something huge, she's asking for you to do something not to lose her, as you have already noticed you could (lose her).
So think about what she likes, and buy something or better yet, make something with your own hands, unless that would take too long, timing is more important, do it soon.
Take into consideration that taking this step could get you into a dynamic of doing nice stuff for her out of guilt, and you don't want that, it is sometimes unavoidable to patch things up as a last resort, but you really don't want your relationship with this person or anyone to fall into a break and patch, break and patch dynamic.
You could gain some emotional terrain and space for yourself if while you do something nice for her, you also remind her you will keep having arguments because you like having arguments with her and wouldn't do it with anyone else because you care about her opinion better than anyone else, like you already told her, only you are now reminding her while being nice, not in the middle of a fight.
If after saying that she tries to pick up the last pending argument, change the topic at any cost, remind her about the present you just gave her, show her how to use it or, if you gave her something she already knows how to use, ask her to show you how to use it, this way you are now doing something she's more confident with than you are.
If she likes to paint, give her a nice set of brushes and pastel colors and ask her to make a drawing of you two or something that represents your friendship.
If she likes to read, give her a book and ask her to tell you the story later.
If she likes tunning cars, well, you get the idea.
The part where you suggest to her what to do with what you just gifted her is not necessary, that's just an idea on how to avoid going back to an argument in a moment when you don't want an argument at all because you are patching things up, but it is important that she understands that you are going to remain firm in your positions and/or beliefs, so your friendship has an opportunity to last.