Marking Anchor Point Set Distance on a Curved Path?

are those fashion patterns?

here's another way that yields accurate results

make a copy of the segment you want to add the 3" notch to (or just make a cut the 3" starting anchor then re join when done)

I made a copy of it and added an arrowhead just to note direction and where the segment ends

We're going to use an Art brush, so create a 3.25" horizontal line and add a vertical notch 3" from from the start.

Select both the 3.25" line and the vertical notch and created a new Art Brush. On the Scale Options, choose Stretch Between Guides and enter 3" Start and 3.25" end. Click OK.

Select your segment and apply the newly created brush

Go to Object->Expand Appearance Menu

Notice the path got expanded into 3 objects, your original segment, the notch and the arrowhead

the Notch is created at the 3" mark and an anchor point was added to the segment. I just cut it with the Scissors Tool to measure it.

By the way, there is a Measure tool in Tools panel. You can drag it along a path to get an idea of the length. It's not perfectly accurate because you are dragging manually along a curved path,  but can reassure you that your measurements are not completely off.

Leveraging off of James Bugge's clever solution.

Select the path. The line weight doesn't matter.

Assuming that you want to add a new anchor point that is 3 inches from the upper left corner of the path, turn on Dashed Line in the Stroke panel. Make sure that Aligns Dashes to Corners is selected. Illustrator makes it easy to convert measurement systems. If you type 6 inches for the dash and 3 inches for the gap, and your units happen to be points, Illustrator will do the conversion for you once you hit the Tab key or click elsewhere to get your cursor out of the cell where the number is typed.

If you enter the numbers shown below, the dash at the corner will be 3 inches (216 points), all other dashes will be 6 inches (432 points), and the gaps will be 3 inches.

If you want the anchor point at a different position, change the dash and gap dimensions accordingly.

Try the plugin VectorScribe. It has a measure tool that measures along a path.

Or try the plugin Inkscribe. It allows you to move points and snap to increments.

I'm not sure if the plugin CADtools allows you to set points at custom distances.

You mention a "curved" path, but the illustration shows a path made of two angled straight lines. Please clarify if you want to add an anchor point on a curved path or a straight path.