Could a simple majority of Senators vote to make the conviction vote a secret ballot?
No, a secret ballot would require changes to the rules and rules changes require a two-thirds of Senators to agree (or unanimous consent).
There appears to be no way that Senators could avoid their names not being known whether the action be a rule change or other.
RULES OF THE SENATE, Rule XXII 2.
... except on a measure or motion to amend the Senate rules, in which case the necessary affirmative vote shall be two-thirds of the Senators present and voting...
To conduct a secret ballot, it would be necessary to submit a "Motion to Suspend the Rules" for, in particular, rules XIV, XX, and XXIII, of RULES OF PROCEDURE AND PRACTICE IN THE SENATE WHEN SITTING ON IMPEACHMENT TRIALS.1
XIV. The Secretary of the Senate shall record the proceedings in cases of impeachment as in the case of legislative proceedings, and the same shall be reported in the same manner as the legislative proceedings of the Senate.
XX. At all times while the Senate is sitting upon the trial of an impeachment the doors of the Senate shall be kept open, unless the Senate shall direct the doors to be closed while deliberating upon its decisions. A motion to close the doors may be acted upon without objection, or, if objection is heard, the motion shall be voted on without debate by the yeas and nays, which shall be entered on the record.
XXIII. An article of impeachment shall not be divisible for the purpose of voting thereon at any time during the trial. Once voting has commenced on an article of impeachment, voting shall be continued until voting has been completed on all articles of impeachment unless the Senate adjourns for a period not to exceed one day or adjourns sine die. On the final question whether the impeachment is sustained, the yeas and nays shall be taken on each article of impeachment separately; and if the impeachment shall not, upon any of the articles presented, be sustained by the votes of two-thirds of the Members present, a judgment of acquittal shall be entered; but if the person impeached shall be convicted upon any such article by the votes of two-thirds of the Members present, the Senate may proceed to the consideration of such other matters as may be determined to be appropriate prior to pronouncing judgment. Upon pronouncing judgment, a certified copy of such judgment shall be deposited in the office of the Secretary of State. A motion to reconsider the vote by which any article of impeachment is sustained or rejected shall not be in order.
Form of putting the question on each article of impeachment
The Presiding Officer shall first state the question; thereafter
each Senator, as his name is called, shall rise in his place and answer: guilty or not guilty.
Rule XXIII requires that the "yeas and nays shall be taken". Rule XIV requires that the "yeas and nays" be "reported in the same manner as the legislative proceedings of the Senate".
A motion to suspend the rules could be used to "close the doors" for the vote on the articles of impeachment and to prevent reporting of the "yeas and nays".
A motion to suspend rules VII, XX, and XXIV was voted during the impeachment of President William Jefferson Clinton. The motion failed due to, "Two-thirds of the Senators voting, and a quorum being present, not having voted in the affirmative, the motion is rejected." 2
Assuming the rules were suspended, the individual votes for "guilty" or "not guilty" would not be reported; however, the individual votes to suspend the rules to allow a secret vote would be reported, unless the rules were suspended by unanimous consent.
It is possible for Senators to be "excused" from voting on the articles of impeachment. A Senator may be excused by a simple majority or unanimous consent.3 The names of the Senators, so excused, would be known. If enough Senators of one party were excused, that would change the political balance for the votes on the articles of impeachment.
1 PROCEDURE AND GUIDELINES FOR IMPEACHMENT TRIALS IN THE UNITED STATES SENATE, p.p. 4-6.
2 PROCEEDINGS OF THE UNITED STATES SENATE IN THE IMPEACHMENT TRIAL OF PRESIDENT WILLIAM JEFFERSON CLINTON, VOLUME II: FLOOR TRIAL PROCEEDINGS, p.p. 1496-1497.
3 PROCEDURE ..., op. cit., p.p. 77-78.