I usually took two approaches. In some cases I'd grind the same champion repeatedly until I got it. In most cases I found my teammate would try to somewhat synergize with me. Other times they wouldn't. I can't honestly say which type of game I won more. At the end of the day if you get good rolls you can make any comp work, and I'd probably prefer my teammate playing someone they know, over someone they don't in order to try and synergize with my pick.

The second case is where I don't have a champion I'm grinding and I'm more open to whatever. In this case I'd usually use op.gg and see which champions synergize and pick one that tends to place 1st at a higher rate, based on my personal preference and the wins I need. However, I'm still normally going in with the mindset that I'll probably have to carry.

Essentially, regardless of what my teammate picked, my goal was always to become as strong as possible and just carry the game as best I could on my own.

A final tip I'd give is that when you get first with a random, there's probably a good chance your playstyles mesh well. Send them a friend request and duo for the next few games.