Periodically one GFCI in the panel trips

This particular circuit runs an outside outlet, one light in the basement and it seems to work 2 switches and a plug in the family room

The problem with the 2 switches in the family room at opposite ends are suppose to turn the plug off and on as a 3 way but it is not wired to work exactly right, could that be the problem?

From one end( switch A), if a light is plugged in to the plug socket controlled by switch A and B, you can turn it off and on, if its on, you can turn it off from the other end switch (B), and on again, but if the switch at end (B) is left in the off position you can not turn on again by flipping (A) again, which is what it should do as a 3 way, if I understand 3 way switching correctly.

Could this botch wired 3 way cause the GFCI to pop off and on, I didn't notice anything else wired to that circuit other than what I listed and one hall light.

I doubt the miswired 3 way is the problem. It is also possible one of the switches has gone bad which could be causing both problems. I am not going to try to sort out the 3 way switches over the net. Check which wires go to each switch and the outlet. You have a pair of wires that is the feed. You must have at least 2 wires going to the outlet plus a 3 conductor with ground connecting the switches. There are several different ways to make then work. The switches are double pole. The contact at the bottom is switched back and forth between the 2 at the top. See if you can draw a diagram where it would work right. If you can't, check a reference work on how 3 way switches work. If it is wired that way and still doesn't work, disconnect the switches and see if they work.

Once you have the switches working, then if the GFIC problem continues, you can work on it. You might pull the outside outlet and clean it up. In damp weather cob webs could trip it.