Meaning of 泡­­­­­­­­­­­­­­­­­­­­­­­­ I first encountered this word in the brand of a kid's drink, O泡.  When I looked it up I saw that it meant something like "bubble", and was (ironically for the name of a drink) the classifier for urine.

Then I saw on TV someone offer 一杯泡水 to a guest (It might have been 一泡杯水 but I read the subtitles and 泡 was definitely there).

So what is 泡水? Bubble water? Google images suggests it's water with slices of lemon/lime.

And while we're on this character, how does one use it as a classifier for urine, perhaps someone could use it in a sentence ("I'm going to do a trickle of urine?").
You are right in saying that, in general, 泡 means a bubble/bubbles. 

When it comes to drinks, 泡水 is actually a verbal phrase, usually preceded by a noun. It refers to water being mixed with other ingredients to create some sort of makeshift drink. For example 柠檬泡水 = lemon water, which is essentially lemon slices submerged in water. 

Another usage of this verbal phrase simply refers to dipping oneself in water. 我去游泳池里泡一下水. (I am going to hop in the swimming pool for a while).

一杯泡水 sounds very strange to me. It could be regional slang for 气水 (soft drink/soda) or it could be the result of making a makdeshift drink. You will need to show me the ad so I can understand the context. 

As a noun, 泡水 is an alternative phrase for 羊水, or the amniotic fluid that protects a foetus in the mother's womb. It's of course commonly referred to as simply water.

As a classifier, it's not just used for urine, but quite often for excrement as well. The following are very idiomatic:


撒一泡尿 (to urinate) 
拉一泡屎 (to defecate)