What component(s) of instant coffee make it deliquescent?
The behavior strongly depends on the temperature and humidity in which the coffee was kept and on the specific type of coffee.

From Tea and coffee powders, M. Huang, M. Zhang, in Handbook of Food Powders, 2013: 


  Being hygroscopic, instant coffee particles are susceptible to the action of moisture – that is, they absorb moisture from the air. If the moisture content increases to 7–8%, the powder or granules can become a paste or solid mass (Clarke, 1987).


Clarke (R.J. Clarke, in Encyclopedia of Food Sciences and Nutrition (Second Edition), 2003) writes: 


  Instant coffee is relatively hygroscopic, easily picking up moisture from the atmosphere and caking at about 7–8% w/w moisture content. For example, to keep instant coffee below 5% (dry basis) moisture content, the relative humidity of the air with which it is in contact must be below 35–40%, though the precise value depends upon the nature of the instant coffee in question, primarily due to differences in porosity of the particular particles made. It is necessary, therefore, that jars of instant coffee be well sealed prior to sale.


My answer is admittedly incomplete. As pointed out in comments to the post and in the above quotes, porosity (characterized by e.g. the mean particle surface-to-volume ratio) influences condensation, but so do the intrinsic chemical properties of the particle surface. 

The chapter by Huang and Zhang referenced above provides the following table (Table 20.1) detailing the composition of instant coffee powders:


  Constituent   Content(% dry weight basis)  
  
  Minerals                  7.6–14.6  
  
  Reducing sugars (glucose) 3.2–13.1 
  
  Galactomannan             2.4–10.5  
  
  Low molecular organic acids   12  
  
  Brown pigments             15–28  
  
  Caffeine                   2.5–5.4 
  
  Trigonelline               1.56–2.65    
  
  Source: Belitz et al. (2009).


Simple sugars are often hygroscopic as may be organic acid compounds, and caffeine. In particular the presence of glucose (part of "invert sugar") could account for much of the absorption of water. In addition, minerals such as various potassium salts (which make up much of the mineral content of processed coffee beans) are known to be deliquescent.

The Belitz reference appears to be: Belitz et al. (2009). Coffee, tea, cocoa. In Food Chemistry (4th Edition), ed. H.-D. Belitz, W. Grosch, and P. Schieberle, 938-970. Germany: Springer. However haven't accessed the original pubs due to paywalls.