What is the role of water in the reaction mixture during synthesis of bromobutane?

Well it may have two roles depending on the water you are talking about. The bulk of the water in this organic reaction is the solvent. Butanol, sulfuric acid, and sodium bromide are all soluble in water and so the reaction is able to take place. During the reaction butanol is protonated by the acid to turn the hydroxide in to a positively charged water group, the water takes its electrons and acts as a leaving group.

The synthesis you gave is obviously a quick, cheap and practical way to generate hydrogen bromide fron sodium bromide in water and then react it with n-butanol to give 1-bromobutane, all in one vessel.

Sodium bromide is not soluble in butanol, so obviously the water serves as a solvent...although in a mixture like this, classification of any reactant is nec essarily a little ambiguous, as then all serve at least one role.