I know how to increase the stroke weight in Illustrator, but I have custom stroke weight points on an object and I'd like to make the whole object's stroke thicker without losing the customized weight points.
Is this possible?
Just increase the stroke weight.
Custom width points scale when the weight of the stroke changes.
If you mean you want width points to remain constant while increasing the stroke weight - that's illogical and not possible since width points are the determining factor of the stroke weight when used. If you want to only adjust part of the stroke to change weight, you need to adjust width markers, not the overall stroke weight.
If you go to Illustrator > Preferences > General and check Scale & Stroke Effects, it will scale the strokes with the rest of the object