The gemara says that Persia thought that it would have better luck than Rome since Rome was responsible for the destruction of the Beis Hamikdash whereas Persia was responsible for the construction - the permission to build it was given by daryavesh (son of esther and achashveirosh). Tosafos asks why they didn't use this as their claim to their contribution and instead they used a claim of everything we did, we did for the Jews to study Torah, similar to the claim of Rome. Tosafos explains that they were looking for a claim that would describe their accomplishments throughout the generation, therefore the building of the beis hamikdash wasn't sufficient since that was a one time thing.
This can be better understood based on the Brisker Rav at the beginning of Breishis who asks what these nations were thinking by being able to pull the wool over G-d's eyes. He explains that they were indeed telling the truth - everything they did was for the Jews because everything that is done in the world is for the Jews. בשביל ישראל שנקראו ראשית. However, although ultimately it was for the Jews, they didn't do it for the Jews and therefore had no claim. Based on this we can add that the claim of permission to build the beis hamikdash was that just as they did that for the good of the Jews, so too everything else that they did was also really for the Jews. Persia felt that their claim would be more believable since they could support everything they did with the fact that they gave permission to build the Beis Hamikdash. Their mistake was that even that wasn't entirely for the benefit of the Jews. The gemara in Rosh Hashana 4a explains that daryavesh insisted that the foundation be made of wood rather than stone so that he can destroy it if the Jews rebel. Even when he gave permission to build the beis hamikdash, it wasn't whole heartedly and therefore was not a good indication that the other contributions were also for the sake of the Jews.