The ability to be my authentic self, and mutual trust.
I'm pretty introverted and most social interactions are draining for me. My few closest friends are those where I don't have to put up even the smallest emotional facade, and therefore can genuinely enjoy relaxing and spending time with on a primal level rather than feeling worn out by it.
I also want my closest friends to be people I can have meaningful and personal interactions with. Sometimes of course you just hang out and play some games and have fun, but if the conversation gets deep I need to feel safe that this happens from a place of mutual respect and that nothing said will get somehow used against me down the line.
Probably the same things people look for in a partner, to be honest.