One interesting, recent (2015), paper which has partially looked at this is Candidate race, partisanship, and political participation: when do black candidates increase black turnout?, by Fairdosi, A. S., & Rogowski, J. C. The paper uses turnout data from the 2010 Cooperative Congressional Election Study to analyse how black candidates affected black turnout in that year's House elections. They chose 2010 as "due to a paucity of black Republican candidates, existing
research has been unable to identify whether increased participation occurs irrespective of the candidate’s partisanship".
Their findings, in their own words:
Consistent with previous research in American politics, we find that
black citizens— Democrats, Independents, and Republicans alike—are
more likely to turn out to vote when a black Democratic candidate
appears on the ballot. However, the same does not apply for black
Republican candidates; black voters are not mobilized by the presence
of a black candidate when that candidate is Republican.
The paper points out that black citizens overwhelmingly vote for the Democratic party, and as a result, it is perhaps possible to hypothesise about how Democratic turnout as a whole may be affected by black Republican candidates. Instead, I have decided to adopt a portion of their methodology, applying it to the 2018 CCES, and extending the sample to all respondents, not just those identifying as black. I've looked at the self-reported turnout rates in each congressional district, and then looked at whether the turnout rates in the districts contested by black Republican candidates in 2018 differ substantially from the overall sample.
There were, to the best of my knowledge, fifteen congressional districts which fielded a black Republican candidate in 2018. These were:
Aja Smith (CA-41)
Casper Stockham (CO-1)
Charlotte Bergmann (TN-9)
Craig Cameron (IL-7)
Herman West (GA-2)
Jeff Dove (VA-11)
Jimmy Lee Tillman (IL-1)
Jineea Butler (NY-13)
Joe Profit (GA-4)
Liz Matory (MD-2)
Mia Love (UT-4)
Tim Westly (TX-15)
Virginia Fuller (FL-5)
Will Hurd (TX-23)
Willie Billups (TX-33)
Below are boxplots showing the distribution of turnout figures according to the CCES for these districts, and those of the election as a whole. Interestingly, we see a slight decrease in mean turnout across all categories, as well as slight changes in skew. A Kolmogorov-Smirnoff test performed on all three variables finds p-values of ~0.175, ~0.285, and ~0.607, so there is insufficient evidence to reject the hypothesis that congressional districts which fielded black Republican candidates do not exhibit exceptional turnout patterns - i.e. there is insufficient evidence to conclude that black Republican candidates increase Democratic turnout.