To directly answer your question: here's an oft-cited work from 1999 that showed that the gender in the word problems can influence performance through stereotype threat (although it did not account for the entire achievement gap in gender): Walsh, M., Hickey, C., & Duffy, J. (1999). Influence of Item Content and Stereotype Situation on Gender Differences in Mathematical Problem Solving. Sex Roles, 41(3–4), 219–240. http://doi.org/10.1023/A:1018854212358
If you want to complexify the question and consider a less binary concept of gender: Indigo Esmonde's work would be a great starting point. Here's a paper to get you started:
which I think eventually became:
Esmonde, I. (2011). SNIPS AND SNAILS AND PUPPY DOGS’TAILS: GENDERISM AND MATHEMATICS EDUCATION. For the Learning of Mathematics, 31(2), 27–31.
but the version I linked to above is longer and more comprehensive.
On a similar question: one might wonder if you should include queer culture and queer families in math problems, and here's one author who discusses what that might entail (as well as how to move beyond that approach):
Rands, K. (2009). Mathematical Inqu[ee]ry: beyond “Add-Queers-and-Stir” elementary mathematics education. Sex Education, 9(2), 181–191. http://doi.org/10.1080/14681810902829646
From my own experience: I know for myself, when I write math problems for middle school students, when putting names and/or genders in the problems, I try to make a point of once in a while having boys do "girl things" and girls doing "boy things", not necessarily designing problems around that, but just being aware of it as I write the problems and occasionally switching it up a bit. I also occasionally when teaching math bring up social justice issues that are currently in the media and ask students to consider them mathematically, so that's another thing to think about - might, say, using math to tackle issues of gender equality have more of an impact in the classroom than just putting women in math problems...?