Faster than Light missiles do not go back in time; they simply arrive at their target faster than a laser beam would, based on their "warp factor" or whatever. They are also required if you universe has faster than light drives on spaceships; you'll need something which can reach a target travelling at superluminal speeds, unless ALL battles take place in "realspace."
Why would I want one? As mentioned in the comments above, lasers still have travel time, and at long enough distances, the target will have moved by the time your sensors give you a location, and even by the time the laser "hits" the target.
FTL rockets don't fix the laser time lag problem completely because they will still have some finite travel time. But they have one major advantage that laser beams do not:
They carry their sensors with them.
Lasers have the problem that, the farther you are from your target, the more round-trip delay exists between your sensor platforms, your aiming accuracy, your computational power to predict paths, their actual course, etc, etc. There's lots of error, but that's usually mitigated by getting closer to your target and ... well, lasers are cheap to fire.
A missile, however, can carry onboard sensors, Which means you don't need advanced course prediction software or high accuracy turrets. You don't even need to worry about sensor delay or gravitational distortion effects. Missiles can constantly course correct to track a locked target, and the missile's accuracy always ends up as "point blank range," because each second it's travelling it gets a little bit closer, and it's sensor round trip time goes down.
Now, either a FTL rocket can scan outside it's drive bubble, or it'll need to "drop out of warp" near the target, re-aquire it, and then attack. But either way, it's way more useful than a laser.
For your example:
With a laser, if you are aiming at a ship five light-minutes away, you need to compensate for the five minutes of travel which happened while your sensor return was coming back, and the five minute travel time for your beam to strike the target. You'll need to predict the ship's course up to 10 minutes in the future. Any sort of "evasive maneuver" would end up throwing your lock, making it nearly impossible to actually hit the target at range, because your data is ALWAYS five minutes behind.
With a FTL rocket travelling at C, you set the rocket to the range of the target (five light minutes) and then fire it in the general direction you predict the target will travel. It will jump to warp and five minutes later drop out and reaquire it's target. It will then make it's own course prediction and jump to warp again, and again, getting gradually closer and closer to the target until it's too close to travel in warp. It will then use conventional engines to hit the target.
Distance, round trip time, gravity wells, all of these are massive sources of error for lasers that are nearly impossible to compute away, but problems which disappear when your weapon can jump to light speed and then correct it's own course, which a laser can never do.