If I want to be a Jew, can I ask the rabbi to give me any name of my choice from the Tanakh?
Examples of converts having different names, are plentiful. In the old rabbinic literature we find converts bearing names such as Onkelos (BT Meg. 3a), Judah (Mishnah Yadaim 4:4 and BT MK 9a) Samuel (BT Yeb. 101b) Yochanan (Midrash PR 14) among other, both Hebrew and Greek names. Examples post Talmud are found as well; one notable one was a contemporary of Maimonides, Ovadiah.
In conclusion, a convert could either keep his current name OR choose whatever Jewish name he likes from the Tanach or elsewhere.