The Great Writ is a relic of the Common Law that has widespread application – it requires a person who has another person in his custody, whether legally or illegally, to bring that person before the court issuing the writ so that his just cause may be tried. It’s the legal ground on which the Guantanamo prisoners are largely depending to seek a trial, for example, and any prisoner seeking a review of court proceedings that found him guilty but were arguably not conducted kosher may, subject to some special rules, use it to bring the arguments about why his trial wasn’t fair before an appellate court. (It’s different than an appeal – it’s legally a whole new judicial action.)
I cannot see how Habeas Corpus could be applied to the circumstances surrounding the election of 1864 – but our present honorable Chief Justice wrote an entire book, largely about the writ of habeas corpus, with his starting premise the circumstances surrounding Abraham Lincoln extralegally suspending the writ as applied to the Baltimore area in 1863, so it’s intriguing you’d bring it up.