First of all, blindness and sickness were widely thought of in ancient times as the consequence of sin. Forgiveness of that sin - the granting of mercy - was the way back to good health. Often when we are told that Jesus cured a person, this is accompanied by his forgiveness of the person's sins and an injunction to sin no more.
These are essentially different accounts of the one event. Knowing this will help us look for the scriptural basis, by finding the reason for the reference to Jesus as 'Son of David' in just one event, rather than many. Nearly all New Testament scholars1 recognise that Mark's Gospel was used by the authors of Matthew and Luke in writing their gospels, so I will look in this earliest account for the scriptural basis.
Mark tells us Jesus leaves Jericho, accompanied by a multitude, The blind man, Bartimaeus, sitting on the roadside, asked Jesus for mercy, twice calling him 'Son of David', and many of those with Jesus told him to hold his peace.
The account in Luke 18:35-39 is almost identical to that of Mark, except that it omits the name of the blind man, Bartimaeus. So there is no doubt this is the same event.
For all its differences, Matthew 20:29-31 is also the same account. Once again, Jesus leaves Jericho, accompanied by a multitude. This time there are two blind men by the side of the road and they say once, in unison, "Have mercy on us, O Lord, thou Son of David." Once again, the people rebuke the men, telling them to hold their peace. This time, Matthew tells us Jesus touched their eyes as part of the healing process.
Raymond E. Brown says in An Introduction to the New Testament, page 181, "The story of the two blind men (9:27-31) is peculiar to Matt; but it resembles closely the healing of the two blind men in Matt 20:29-34." Brown goes on to say Matthew "has a penchant for doubles." As in Matthew 20:29-34, the story in Matthew 9:27-31 has two blind men who call Jesus the Son of David and beg him for mercy. Once again, Jesus heals their blindness by touching their eyes. That this is an adaption of the account in Matthew 20:29-34 is further verified by the fact that these are the only two occasions in Matthew in which we are told that Jesus touched the eyes of the blind in order to cure them.
It may be noticed in Mark's Gospel that Jesus is only ever referred to as the "Son of God" by outsiders. For example in Mark 5:1-20, the demons, as outsiders, call Jesus “Son of the most high God,” but Jesus only ever refers to himself as the "Son of Man" and even Peter, when asked who Jesus is, calls him the Messiah (anointed one). Of course this changes in the gospels written decades later, but the author of Mark seems at pains to avoid a charge of blasphemy from either the Romans or the Jews. So Mark resolves this by having the blind man call Jesus "Son of David," but we know what is really meant.
1Adam Winn (The Purpose of Mark's Gospel, page 1) says "the theory of Markan priority is one of the few that has reached a high level of consensus among New Testament interpreters."