Hizkuni to that verse (9:35) writes that it means "in Moshe's prophecy", and that that is what the term ביד משה always means. Radak also writes in his commentary to Hoshea (12:11). Indeed, the term is used dozens of times in the context of prophecies conveyed by Moshe (e.g. Exodus 35:29). The term is not unique to Moshe, and is used for other prophets as well.
- I Kings (12:15, 14:18, and 15:29) uses the term for prophecies conveyed through Ahiya HaShiloni, as does II Chronicles (10:15).
- I Kings (16:7 and 16:12) uses the term for prophecies conveyed through Yehu.
- I Kings (16:34) uses the term to refer to prophecies conveyed through Yehoshua.
- I Kings (17:16) uses the term for prophecies conveyed through Eliyahu, as does II Kings (9:36 and 10:10).
- II Kings (14:25) uses the term for communication through Yonah ben Amitai.
- Isaiah (20:2) uses the term for communication through Yeshaya.
- Jeremiah (37:2 and 50:1) uses the term for communication through Yirmiyahu.
- Hagai (1:1, 1:3, and 2:1) uses the term for prophecy through Hagai.
- Malachi (1:1) uses it to refer to prophecy through Malakhi.
- I Chronicles (11:3) uses it for prophecy through Samuel.
- I Samuel (28:15) uses the expression ביד to refer to communication by God through prophets in general, as do II Kings (21:10 and 24:2), Ezekiel (37:19), Zechariah (7:7 and 7:12), Daniel (9:10), Ezra (9:11), Nehemia (9:28 and 9:30), and II Chronicles (29:25). Furthermore, II Kings (17:13 and 17:23) uses the expression for God's communication through all the prophets.
Given that the expression is not unique to Moshe, there is no reason to suppose that it refers to communication through hands. Note also II Kings (4:27) in which the term ביד is used for God acting through someone, outside the context of prophecy (although in that case it could be claimed that it refers to actions committed with hands), and note also II Kings (19:23) and Psalms (77:21). Note also Hoshea (12:11) with Radak's commentary, and Zecharia (9:16) with Radak's commentary, in which ביד is used outsides the context of prophecy, or hands. Similarly, note Isaiah (64:6) with the commentary of Metsudat Tsion in which ביד is used to mean 'through' outside the context of prophecy (and of hands).
Rather based on all the above textual evidence, it appears that Hizkuni is correct that ביד משה means "through Moshe." This seems similar to the expression על ידי in Jeremiah (18:21), Ezekiel (35:5), Psalms (63:11), and Ezra (3:10) which is quite common in Mishnaic Hebrew that also means 'through'.
For a Midrashic interpretation of one instance of ביד משה, see Rashi to Numbers (17:5) who associates hand with Moshe's tsaraat which he had on his hand, suggesting that those like Korah (mentioned in that verse) who disagree with the legitimate priests, will be stricken with tsaraat.
Another derash is that of Paneah Raza (Parashat Tsav) who suggests that ביד משה (in the context of Leviticus 8:36) suggests that Moshe acted with all the strength in his hands to serve God.