There is a מנהג that women do not do מלאכה while the נרות חנוכה are lit [1]. The reason for this is so that they should have a reminder that it is אסור to use the light of the candles [2]. Additionally, in order that they shouldn't be distracted from remembering the miracle they therefore make that half-hour into a יו"ט [3].
This rule only applies to women because the נס occurred through them [3]. The מהריל, however, actually brings this מנהג as applying to men as well. Additionally, there are those who say that there is a general איסור-מלאכה on the first and last days (similar to the first and last days of סוכות/פסח) [5].
Practically, the ערוה"ש testifies that we have never heard of most of these מנהגים, to which end our women only refrain from מלאכה during הדלקה [6].
It would seem that today (to my knowledge) this מנהג has been largely forgotten (in some circles). This may be a result of the prevalence of electric lights, which render the entire דין of אסור להשתמש לאורה to be largely obsolete. (No one who has electric lights in their home would really need to rely on a few small candles for illumination.) To that end, a reminder that using the חנוכה candles is אסור is, to a significant degree, redundant, as few people would even have a הוה אמינא of doing so in the first place.
Addendum
In order to get a clearer demographic picture of where this מנהג is prevalent and where not, I have created a survey with pertinent questions that can be filled out here:
https://forms.gle/GcWEsgcuNgc84cno6
[1] שו"ע או"ח תרע:א. This only applies to the time that they need to be lit, which is half an hour. After that there is no איסור (מ"ב שם ס"ק ד).
[2] באר הגולה שם אות ג
[3] לבוש שם. Perhaps men don't need as much of a reminder as women because men are the ones that are actually lighting.
[3] מ"א שם ס"ק א
[5] הלכות חנוכה אות יא בהגה. See note 5 in the מכון ירושלים edition.
[6] שם סעיף ח. The precise language of the ערוה"ש is "בעת הדלקה". It is unclear if the ערוה"ש means so long as the candles are lit (as the other פוסקים say) or only while the candles are in the process of being lit.
