$B(\chi), L'(1,\chi)/L(1,\chi),\dotsc$ Let $\chi$ be a primitive Dirichlet character of modulus $q>1$. Write, as is customary, $B(\chi)$ for the constant in the expression
$$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} = B(\chi) + \sum_\rho \left(\frac{1}{s-\rho} + \frac{1}{\rho}\right),$$
where $\Lambda(s,\chi)$ is a completed Dirichlet $L$-function and $\sum_\rho$ is a sum over its zeros. Obviously,
$B(\chi) = \Lambda'(0,\chi)/\Lambda(0,\chi)$. Since
$$\frac{L'(s,\chi)}{L(s,\chi)} = \frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} - \frac{1}{2} \frac{\Gamma'((s+\kappa)/2)}{\Gamma((s+\kappa)/2)} - \frac{1}{2} \log \frac{q}{\pi},$$
where $\kappa = [\chi(-1)=-1]$, we see that $$B(\chi) = b(\chi) - \frac{\gamma}{2} - \kappa \log 2 + \frac{1}{2} \log \frac{q}{\pi},$$ where $b(\chi)$ is the constant term in the Laurent expansion of $L'(s,\chi)/L(s,\chi)$ around $s=0$. We can easily show that 
$$b(\chi) = \log \frac{2\pi}{q} + \gamma - \frac{L'(1,\overline{\chi})}{L(1,\overline{\chi})}$$ by taking logarithms on both sides of the functional equation. We thus obtain that
$$B(\chi) = \frac{1}{2} \log \frac{4^{1-\kappa} \pi}{q} + \frac{\gamma}{2} - \frac{L'(1,\overline{\chi})}{L(1,\overline{\chi})}.$$
It seems clear to me that this expression for $B(\chi)$ must be (very) classical. Now, looking in Montgomery-Vaughan for something else, I see that, in section 10.3, it states that "The constant $B(\chi)$... was long considered to be mysterious; the simple formula (10.39) for it [namely, the expression for $B(\chi)$ right here] is due to Vorhauer (2006)." Here Vorhauer (2006) is an unpublished preprint (not accessible online). I'd gladly give credit where credit is due, but I can't help thinking that this expression must have been known long before 2006. Does anybody have an earlier reference?
(And what would be so mysterious about $B(\chi)$? IMHO, it  is just tricky for the same reason that $L'(1,\chi)/L(1,\chi)$, viz., the possibility of a Siegel zero. Or is it just that we don't have an expression for it as nice as the class number formula? (Do we? EDIT: for $\chi$ odd, we do; see Prop. 10.3.5 (due to...?) in Henri Cohen's Number Theory.) On the issue of bounding it, see $|L'(1,\chi)/L(1,\chi)|$.)
 A: I am replying to this question “for  odd, we do; see Prop. 10.3.5 (due to...?) in Henri Cohen's Number Theory”. I would be happy to insert a comment instead, but my MO-reputation is not good enough...
In a paper published in 1989, Kanemitsu wrote that this formula was first published by Berger in 1883. 
Kanemitsu’s paper is entitled “ On evaluation of certain limits in closed form”, pages 459-474, of the volume:
Proceedings of the International Number Theory Conference held at Université Laval, July 5-18, 1987
Ed. by Koninck, Jean M. de / Levesque, Claude
Series:De Gruyter Proceedings in Mathematics. 
Berger’s paper is “sur une sommation des quelques series”, Nova Acta Reg. Soc. Sci. Ups.(3) 12 (1883).
A: When I was a graduate student at the University of Michigan (this would be the mid-1990s), I took an analytic number theory class from Montgomery, from notes that would eventually become his book with Vaughan. I remember learning directly from Montgomery in that class that the real part of $B(\chi)$ could be written in terms of the zeros of $L(s,\chi)$, but that the imaginary part was indeed mysterious.
Perhaps part of our mental block as a discipline was that the usual formula for $\Re B(\chi)$ contained the term $\Re \dfrac{L'}L(1,\chi)$, while Vorhauer's formula for $B(\chi)$ turns out to contain the term $\dfrac{L'}L(1,\overline\chi)$ rather than $\dfrac{L'}L(1,\chi)$. (Note that the formula in your post contains an omission in this regard.)
In any case, given the timing of this information, and the fact that Montgomery is a central figure in classical analytic number theory who is also dedicated to knowing its literature, I am confident that the formula in question is indeed due to Ulrike Vorhauer as noted. I think the correct thing to do is to credit Vorhauer with the formula's discovery and cite the book of Montgomery and Vaughan as the best source we have.
Edited to add: I have checked Davenport's book, and the formula it gives for $B(\chi)$ at the top of page 83 is not the same as Vorhauer's formula (an infinite sum over zeros is still present in Davenport's formula). The quote "can be expressed in terms of the expansion of $L'/L$ in powers of $s$" does not at all imply that Vorhauer's formula was known (for instance, it gives no hint that the distinction between $\chi$ and $\bar\chi$ is relevant); it corresponds only to one of the very first steps in the sketch from the OP. Moreover, Montgomery himself revised Davenport's book; it strains credulity that he, having carefully read page 83 of Davenport, would attribute the formula to someone other than Davenport if that page were a sufficient source for the formula.
It's one thing to say that Davenport and those who preceded him could have derived the formula (that much seems clear). But what evidence we do have points to the conclusion that nobody actually derived Vorhauer's formula until she did. That sort of thing happens all the time. We still give credit to the actual discoverers (Vorhauer, in this case); we don't deem the result "classical" based on our feeling.
Edit 2: Apparently Vorhauer's paper was accepted to Acta Arithmetica, but the publication process stalled at the page-proofs stage.
