When making a bracha with shem u'malchut, one should not have erva (nakedness) exposed. For married women, their hair is considered to be erva. So does a woman who is married have to cover her hair when she makes a bracha?
Many ladies when they go to the Mikva have their hair uncovered when they make a Bracha.
See also Mishna Mesechtas Chala האשה יושבת וקוצה חלתה ערומה - from the Biur it clearly seems like she takes Challa and makes a Bracha in that state.
However Yabia Omer 6:15 says that a lady should have her hair covered even if in Chadrei Chadarim when saying Hashem's name.
While men cannot say blessings in the presence of a tefach of a woman's exposed body-part-which-is-usually-covered (ShA OC 75:1), Arukh HaShulchan OC 75:4 explicitly rejects an opinion that the same applies to women. Rather, women can say blessings in front of an exposed body-part-which-is-usually-covered of another woman. (Even the rejected opinion agreed that a woman can say blessings in front of her own exposed body-parts-which-are-usually-covered.) Thus her exposed hair (or shoulder, or thigh, or breast), even if it is usually covered, would not present a problem for her or any other woman in terms of reciting blessings in the presence of "erva". (I note it is still forbidden for anyone to recite blessings in front of exposed genitalia, including their own.)
There is a separate general obligation of covering the head (eg. with a Kippah) when reciting God's name as an act of respect (ShA OC 91:3). Classical sources do not distinguish between the genders here. While the Ashkenazi custom has largely been to be lenient about this for women, rabbis have struggled to find a basis for this practice (see, eg., Tzitz Eliezer 12:13). Sefardi practice has historically not been as uniform and many contemporary Sefardi rabbis do encourage head-covering for women when saying blessings (see, eg., Yabia Omer OC 6:15 who encourages it particularly for Shemoneh Esrei and Bentching).
Ovadia Yosef holds that women, married and unmarried, ideally cover their heads when they say shem u'malchut, regardless of location or others' presence. In a Yeminite shul, unmarried women will be asked to cover their heads and I have seen unmarried Yeminite women cover their heads to light and bless Chanukah candles.
For those who are interested in the issue of whether women's headcoverings have significance outside of modesty, this looks like a good article, though I personally haven't gotten to it:
Hair Covering for Single Women: A New Reading of Mizrachi Halakhic Rulings. Ilan Fuchs. Nashim: A Journal of Jewish Women's Studies & Gender Issues. Number 23, Spring-Fall 5772-3/2012