Why does Devarda's alloy include copper?
Virtually every analytical chemistry textbook's chapter on anion determination mentions that Devarda's alloy is very brittle and can be easily pulverized in a mortar, unlike $\ce{Al}$ and $\ce{Zn}$ metals.
This is quite handy property as for qualitative analysis of nitrates in a heterogeneous media the reducing phase should be ground to fine powder in order to:


assure micro-scale precision of the added reactant, which is tricky to achieve with aluminium or zinc, as both metals are usually shipped in granules, and the fine metal powders are prone to oxidation, increasing the amount of unwanted oxides (plus, their powders are sticky);
increase the available reaction surface to speed up the analysis and increase its completeness (especially important for isotopic nitrate analysis).


On top of that, German Wikipedia 
also supports both comments by Ivan Neretin


  Galvanic couple, I guess.


and aventurin


  Catalyst, I guess. ;-)


and suggests that copper facilitates catalytical formation of a local electrochemical cell.
A reduction with Devarda's alloy can be carried out in weakly alkaline solution or even in neutral solution, whereas reduction with aluminum often requires a strongly alkaline solution.
At the same pH, the reaction with Devarda's alloy is overall significantly faster than when using zinc or aluminum.

This was experimentally demonstrated by Mertens et al. [1]:


  The superiority of Devarda alloy compared to aluminum dust as a reductant in alkaline medium is illustrated best with the following experiment (see Figure 3).
  Twenty ppm $\ce{NO3-}$ is subjected to reduction, first with aluminum dust and the potential of the ammonia probe is recorded continuously.
  One observes a slowly rising potential.
  On addition of Devarda alloy to this reaction mixture, a potential jump is obtained and the potential stabilizes rapidly (after 5 minutes).
  This shows clearly that the kinetics of the reduction with Devarda alloy are much more rapid than with aluminum dust. 
  
  
  Figure 3. Potential observed with the ammonia probe during the reduction of 20 ppm $\ce{NO3-}$ as a function of time
  From $t = 0$ to A: aluminum dust. At point A, Devarda alloy was added.  


References


Mertens, John.; Van den Winkel, Pierre.; Massart, D. L. Determination of Nitrate in Water with an Ammonia Probe. Anal. Chem. 1975, 47 (3), 522–526. DOI: 10.1021/ac60353a009.