There are a few possibilities:
- Play lane bullies who have few counters. Avoid late-game champs with weak early-game (Nasus), or very specialized champs with lots of hard-counters (Katarina).
- Play your best champ. With a bit of luck, your level on this champ is so much above your average level (i.e. the level of people you play against/with) that the counter doesn't matter.
- Play Jungle. You don't really face the other jungle, so it matters a lot less than in lane. (I heard people say the same thing about Support, but I strongly disagree.)
If possible (playing with friends, or good at diplomacy in soloQ), you can also:
- Play multi-lane champs, so they don't know what lane to counter.
- Let someone else take the burden, and pick-and-swap for them.
Also, I agree with Captain Aishi: playing against your counter is not fun, but it's very useful in the long term, to better understand your champ and learn to play him better, and the game in general too. Learn to play defensively, to just not lose your lane, and wait for teamfights. Remember that if you can be countered, your team also has the opportunity to counter your enemies on other lanes, so they should have an easier time than you, so sometimes just not losing is enough.
And remember, if you lose because you got countered, it's not your jungler's fault, and he has no obligation to baby-sit you. Sometimes it's the good decision, but sometimes not, and you should be able to be self-sufficient in lane.
P.S. : if you queue with someone else, it's the person that owns the lobby and click on "start game" that will be first pick (or can be first pick if you play at only 2 or 3). You can't be first pick if you have been invited / have given the lead to someone else.