To shove something into a tight place, in a less-than-gentle fashion. You can make it fit!
Person one: "Oh, my suitcase is full. I can't fit anything more in it, I don't think!"
Answer one: "You could stog a few more pairs of socks in the corners, I betcha."
Answer two: "Nah, stog 'er in there!"
A ridiculously cheerful song by the pop-rock band wordHanson/word, released widely in 1997, and was a huge hit. Despised by many, but beloved by true wordFanson/words.