What is the role of Congress in determining when habeas corpus can be "suspended?"
The role of Congress in determining when the writ of habeas corpus can be suspended is not at all clear.
The Constitution does not say who can suspend the writ of habeas corpus. All that it does say (in Article I, Section 9, Clause 2) is that
The Privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it.
The Constitution does not say who gets to determine if there is a rebellion or an invasion. (For example, were the 9/11 attacks an invasion?) It does not say who gets to determine whether "the public safety" makes it necessary to suspend this right. In the Civil War, both the President and Congress suspended habeas corpus in one case each. This implies that Congress does have some role in the process.
It is, then, not clear what role Congress plays. Congress appears to have the right to suspend habeas corpus. However, that is clearly (as in Boumediene v. Bush) subject to review by the Supreme Court. It is also not clear if the President or Congress has the greater role to play in determining when habeas corpus may be suspended.