I have a diagonal line that's been subdivided, so it has a point in the middle. I want to move this point so that it vertically aligns with other another point along the global Y axis, but so that, and this is the most important aspect, it still stays perfectly aligned with the rest of the diagonal line segment it lies on.
The shortest distance between two lines in Euclidean space is a linear segment, so this technique is essentially just minimizing the distance along the local Y axis by forcing the diagonal point to be perpendicular to the vertical segment by first constructing the point upon which applying that operation yields the alignment in the global basis.
Below is the series of steps.
Step 0: Identify the points you want to be aligned
Step 1: Extrude the Global-Y aligned point until it intersects with your diagonal line.
Step 2: Use one of the answers from here to move the cursor to the edge intersection How can I add vertices to intersection of two edges?
Step 2.5: You may need to enable the "add mesh: extra objects" addon on your user preferences
Step3: Create a new axis with an orthogonal basis containing the diagonal line, basically just select any two points on the diagonal line, then hit CTRL+ALT+SPACE to create an axis for that line.
Step4: Make sure your pivot is set to 3D cursor, make sure your transform orientation is the newly created axis. Select the diagonal point and hit "s" then press "Y" and then "Y" again to scale along the local Y axis you just created, and then the vertex should now be alinged both with your diagonal line as with as with the global Y of the other point.