It seems awfully strange to mention a very strong power, that if used improperly could destroy American democracy, and not say who can invoke it. Why?
I'd expect that if you asked the framers, they would say that they were pretty clear about who can suspend habeas corpus-- Congress. The only reference to habeas corpus in the Constitution is in Article 1, which establishes the legislative branch, Section 9 (Powers Denied Congress) which reads
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 fact that Congress is denied the ability to pass a law suspending habeas corpus except in cases of rebellion or invasion means that Congress does have the power to suspend habeas in cases of rebellion or invasion (which the President would then have to sign). The Constitution intended to give very few powers to the President alone-- the ability to command the military (though not to declare war), to issue pardons and to make recess appointments without the assent of Congress being exceptions. The President was initially intended to primarily execute the laws passed by Congress.
When Lincoln suspended habeas, he was rebuked by the Supreme Court which found that it was Congress that needed to suspend habeas. Lincoln's reply was primarily a practical rather than a Constitutional answer. In the case of civil war, keeping the government functional so that it can execute all the other laws is more important than following the legal niceties around suspending habeas. Of course, that argument would carry more weight if Lincoln had moved expeditiously to ask Congress to assent to his suspensions.
It is reasonable to agree with Lincoln that the framers didn't consider the practical realities of governing during a civil war while drafting the Constitution. But the framers were pretty clear about their intentions on the suspension of habeas corpus.