Is there a way to look up a match with a specific Summoner in it?

I'm guessing he's one of us who often looks at their friends list and thinks "Gee, I don't remember any of these 'friends' who are online right now. I sure wish I could quickly recall the matches that prompted me to add them. " It's OK bro, we feel your pain haha.

My suggestion? Add a feature to grab a screenshot of your friends' champs during replays, which are then tied, clientside, to their appearance in your friends list somehow. For example: I could add that amazing hyper aggressive Tristana ADC I supported earlier, then go into the replay viewer and grab a screenshot of her landing after a rocket jump, rocks flying into the air, with my shield and ball floating around her. Later, if I ever wonder who XxDrAgOnSlAyErxX is, I click his/her name in my friends list and up pops that epic screengrab, instantly reminding me of the game I played with that player. It would be a great social feature, and you know how everyone loves social features... Think about it Riot!

You mean look for a player in your match? Search your or someone else who was in the match, summoner's name on OP.GG (make sure you search on the server you are on), there you will be shown your match history and can see the names of the people that were in the matches.

You can also instead send a ticket through Player Support and they may help you find it depending on the reason. You should probably only use this if you can't find the game or if it was a long time ago.