The song that made me think of this is Amarillo by Morning, made popular by George Straight, but it's quite common in a lot of country and pop music. In the verse at the end of the second 4 measure phrase there is one extra measure where the song sort of collects itself before moving on to the chorus. I have a friend that I jam with that puts these kind of one measure pauses into like every song we play, so I was wondering if there was a name for this kind of thing so I can know what to call it when he does it.
This is an issue of what we call hypermeter; specify, we can call it a hypermetrical extension. By hypermeter, I mean a metrical structure not at the level of the beat, but at the level of the entire measure.
Try conducting along with "Amarillo"; it's in 4/4 time. After you're used to that, try conducting in 4/4 where every beat is the start of a new measure. It's going to feel really slow, but that's correct: you're actually conducting a quarter of the speed you just did. Starting at 0:20, for instance, you'll conduct one beat every time the bass guitar changes pitch.
Just as 4/4 is a really common meter, a four-measure hypermeter is also very common. They opening hypermetrical phrase is something like this:
1 Amarillo by morn-
3 Up from San Anton'.
With this pattern set up, we expect the four-measure hypermeter to continue. But what happens in the second phrase is there's a fifth measure inserted before the start of the next phrase:
1 Everything that I've
2 got is
3 just what I've got on.
5! When that
1 sun is high...
This fifth measure that you're discussing is the hypermetrical extension. (A less fancy term would just be "phrase extension.")
For the opposite of this process, check out "Every Little Thing She Does is Magic" by The Police. Conduct the four-measure hypermeter at the start; your conducting will match the bass guitar pitches. It may even be helpful to count out the slow "1 2 3 4" as you're conducting. You should notice something a little strange around 0:45! Here they actually shorten the hypermeter, opting instead for a three-measure hypermetrical phrase.
Edit: This "Amarillo by Morning" is a bit more interesting than that: when he returns to the "Amarillo by Morning" text at 0:46, he inserts a section with only three measures in the hypermeter. This means that, zoomed in, there are two "errors" in the hypermeter: one bit is one measure too long, one is one measure too short. But from a global perspective, it results in a normative 16 measures of hypermeter; clever!