it's likely the power port is bad. i see it all the time in my shop. the good news is they are cheap to replace and not that hard on most laptops.
i get em off ebay all the time for less than $10. most of the time they are simply screwed in and easily replaced once you tear down the laptop. one way to tell if this is the case is if you can sometimes wiggle around the charger in the port and get it to light up for a moment. or if when fully plugged in, you can move the jack around a lot. often the plastic insert that holds the jack cracks allowing the plug to move around a lot and not make the connection.
i've also seen older laptops where the ground wire has cracked inside keeping it from connecting right. same solution is to replace the power port. youtube is a great place to find a video tearing down your specific laptop model. i've yet to find a model i could not view a teardown for on youtube
geek squad wants the money so they'll replace the mobo even if it is only a $7 part that needs replacing. it's a risk you take when going in there.