The western border of Texas and New Mexico is quite odd: it runs and weaves back and forth in a seemingly random way without following the natural border of the Rio Grande. How was it drawn the way that it is?

OK. This is a familiar part of the world to me, as I used to make and update street maps of this area, and had to show the rather odd state boundary.

Today's Río Grande has both moved naturally, and has been channelized, in this area. However, the U.S. Supreme Court decided in 1927 that the boundary between Texas and New Mexico in this area was a fixed line located in the channel of the Río Grande as it ran on September 9, 1850. The problem was that no source showing the 1850 location of the river could be found. The Court appointed a special master to survey and monument the proper line, and this was eventually accepted by both states. This article gives the details.

The boundaries of many U.S. states include areas where a river has moved, making part of the state inaccessible by land without going through another state. Kaskaskia, once the capital of Illinois (when it was a territory) is now on the Missouri side of the Mississippi River. Carter Lake, Iowa, an Omaha suburb, is now west of the Missouri River even though it's part of Iowa. The details of state boundaries are given in Boundaries of the United States and the Several States, a US Geological Survey book available as a PDF here, or older versions on archive.org or Google Books. Mark Stein's book How the States Got Their Shapes is a popular book telling the same tales. It's more fun to read, but somehow completely misses any mention of the Washington Meridian.