Is it possible to set the name of an enum option via construction script in BP?
You can’t change enum fields after they are declared. Not in Blueprints neither in C++
Bummer. Oh well, not a crucial feature. Thank you for the info! 
What exactly do you want to do? Making an ENUM readible in UMG?
If so you need to use a switch over ENUM and return the specific Text to this Enum.
I’m setting up configurable weather profiles; you set a large number of parameters to fine-tune a weather profile you want, select a profile (struct variable) from a list of available options (organized with an enum) and click a bool checkbox to save it (which just sets the bool checkbox back to false immediately, similar to the refresh function in the default skybox.)
I was wanting one of the paramters to be a text field which would feed into the enum options, set the enum option name from that text field, then when you save it, it changes the name and clears out the text field; then you’d be able to find your profile from that dropdown, because it would have taken that name.
However since it doesn’t appear to be possible, I’m just setting up “Custom 1, Custom 2,” etc as additional options outside a few template patterns.