Many things came into the tradition after 70 C.E. That is partially because the movement was forced to take a new turn. And really, it doesn't seem as if Paul was all that well preserved. In that, I mean we are missing letters from Paul, we are missing all of the letters to Paul, some of the letters do have interpolations, and others are just a jumbled of possibly multiple letters (2 Corinthians is like that).
Paul also doesn't seem all that important at this early date. For one, his letters weren't preserved in the best way. Acts doesn't even really refer to Paul as an Apostle (which is interesting, because Paul does make a very strong point of being an Apostle, so we know that is an early tradition, yet Acts changes that as well). Acts also, on occasion, directly contradicts Paul, so it wasn't as if the tradition was cemented yet. As a side note, Acts also seems oblivious to the idea that Paul was writing letters, which, from what we see with Paul, was actually a major aspect of his ministry. So the tradition wasn't very firm at all.
As for why would Acts create such a story about Paul being a Roman? It gives Paul a sense of Universalism. Luke-Acts, from the beginning, is trying to make a point of stressing that even though the message started for the Jews, it became a universal message. Paul is the means for the universalism. His Roman citizenship is key here, as it spreads the message not to just Jews and Gentiles, but to even the elite of society. It encompasses everyone. So that is a good reason for Acts to create such an idea, as it fits into the theme of Luke-Acts.
Acts also has a tendency to build up Paul. In Acts, Paul is taught at the feet of Gamaliel. That is quite prestigious. Yet, Paul never mentions such, which is odd since that is something one would assume Paul would mention. Being labeled Roman citizen also builds Paul up, and exaggerates him, as being a Roman citizen was a special status. It was seen as being the elite. There was a prestige with it. And for Paul, it was even more because unlike the centurion he speaks to, he didn't have to pay for such an status, he was born into such. So it builds Paul's reputation, and it builds up the figure of Paul. It gives him more authority.
It also allows the message to be presented to other Roman citizens. We have the Roman centurion as I mentioned above. But it also brings Paul to Rome, and gives him an elite audience. So that universalism comes into play again. The message is for everyone, and since Paul fits in everywhere, he is able to preach the message everywhere.
So creating such an idea, that Paul was a Roman citizen, makes sense when the theme of Luke-Acts is examined. Paul symbolizes this universalism that Luke-Acts really is centered on. More so, it gives Paul more authority. And it makes certain aspects of the narrative more exciting. Paul going to Rome is exciting, when it is in the context of a trial, as in Acts. Paul going to Rome, in order to talk to the churches, is boring, as what Paul states is his plan in Romans.