I didn't actually know the answer to this, so I did a little bit of digging. As with any historical trend it's difficult to pinpoint exactly where something like this starts, but I've found something that seems like a reasonable explanation for how it spread.
So searching by date in Google yields evidence of these buttons on the English web going back to at least 1996. At this point in time the largest provider of personal hosting was GeoCities. In order to improve brand awareness, they required that all free hosting users have a link back to GeoCities somewhere on the page. They helpfully provided default banners for these links at - you guessed it - the dimensions of 88x31 [1].
GeoCities users being very concerned about design, they objected to the generic banners not matching the style of their pages. A small community of button modders popped up to provide more "attractive" alternatives that fit into the same space [2].
What's more, since most people with web pages had GeoCities pages and therefore had to accomodate this banner into their design, it seems likely that people started to make same-sized buttons for their own sites that could appear alongside the GeoCities banner in a nice little grid. One thing led to another and before you knew it, any self-respecting website had a little branded button to offer to anyone who wanted to link back to it. (And the popularity of link exchanges and web rings at the time would only have helped this trend become viral.)
A 1996-1999 collection of micro buttons. Source: http://www.complexify.com/buttons/
---
- http://reocities.com/NapaValley/2022/icons.html
- Unfortunately, in 1998 they pushed this further into a watermark that broke a lot of layouts and couldn't be removed from the page or replaced, upsetting a lot of users. http://news.cnet.com/GeoCitizens-fume-over-watermark/2100-1023_3-212596.html