On his blog Charles Seville mentions Rackham's comments on why Luke sometimes mentions Saul/Paul before Barnabas, and sometimes reverses the order. There is always significance to name order in the NT, and I think the order that Luke chooses in each case fits with what we know from the rest of Acts and from Paul's letters.
The order is not the same throughout and this shows that the two men had roughly the same prominence. This is in agreement with what we learn from Paul (Gal 2:1, 9, 13; 1 Cor 9:6).
Barnabas is always mentioned ahead of Paul until Paul becomes the dominant speaker from Acts 13:9 onwards (see Acts 11:30; 12:25; 13:1; 13:2; 13:7). This suggests that Saul was not prominent prior to the 'first missionary journey', which is consistent with the fact that Luke has little to say about these first ~11 years of his Christian life, and it is also consistent with the fact that we have no letters of Paul from this period. Paul prominence in preaching after Acts 13:9 (see Acts 14:12) explains why he is mentioned first in Acts 13:43; 13:46; 13:50; 15:2; 15:22; 15:35; 15:36.
There are only 3 times after Acts 13:9 when Barnabas is mentioned before Paul. The first occurs in Acts 14:12-14 when the followers of Zeus attempt to offer sacrifice to Barnabas and Paul, having concluded that Barnabas was Zeus. Barnabas is more central to the story than Paul and this explains why he is mentioned first.
The 2 other cases are in connection with Jerusalem (Acts 15:12; 15:25) (see also Acts 11:30; 12:25). This is presumably because Barnabas was better known then Paul in Jerusalem (Acts 4:36-37; 9:27; 11:22; Gal 1-2).
Thus the name order of all 15 cases is explained. The data confirm Paul's relative obscurity in the early years, and Barnabas's prominence in Jerusalem.