Not sure if any of these really qualify as “obscure,” but some suggestions…
Jared Stevens - Fate - definitely not obscure at the time, but perhaps a bit lost to the sands of time and worth revisiting?
Blink - Lee Hyland - before he was a Mother Panic cameo, he was a Dwayne McDuffie character in Legends of the Dark Knight#164-167.
Human Bomb - Michael Taylor - 2012 “modernization” of the original Freedom Fighter.
Human Bomb - Roy Lincoln - Freedom Fighters isn’t on DCUI yet, though it’s been requested many times, but we could read the Crisis on Earth-X arc in All-Star Squadron#31-35
The Dark Mansion of Forbidden Love - I think each issue it a self-contained story, so we could do one or a handful. I think the recent series of the same title is a rough amalgamation and remix of the first four, before the title changed.
Green Man - maybe not so obscure now the Geoff Johns has used him so heavily, but it’s always fun to follow a fallen GL (right?). There are huge gaps in the Green Lantern run on DCUI, so we can’t read his first appearance, but he’s got a short backup story in #172, and his post-Corps discovery in the tail end of The Omega Men (#30-38) might be a good bite-sized read.
What about a spotty Bloodlines arc? Other than Hitman, most only appeared in their respective Annuals, though a handful had longer arcs outside the event. There’s a bunch of DCUI induced gaps, but that leaves enough characters to do a read-a-long of the rest, I think. (Though sadly, no conclusion for comprehensive arc.) In order (I think): Layla, Edge, Joe Public (missing: Shadow of the Bat Annual #1), Argus, Anima, Myriad, Nightblade (missing: Green Lantern Annual #2), Ballistic (missing: Batman Annual #17), Lionheart (missing: Justice League International Annual #4, so never gonna happen), Razorsharp, Loose Cannon, Jamm, Hook (missing: Green Arrow Annual 36), Geist, Terrorsmith (missing: Justice League America Annual #7), Sparx, Mongrel, Gunfire, Prism (missing: Eclipso Annual #1), some guy named Tommy, Cardinal Sin & Samaritan (missing: Legends of the Dark Knight Annual #3), Chimera, aaaaaaand Pax. So that’s 15 of 25 books available (60%), but 14 pretty obscure characters?
Amazing-Man - Will Everett - back to All-Star Squadron, we could do his intro run (#23-27), or better yet, take up the Real American arc in #38-40.
Some ideas that come to mind – I haven’t read most of these, but ones that have caught my eye in the past.
1st Issue Special – A series of one-shots with more obscure characters that were meant to be the start of new series cancelled after the DC Implosion. Many of these characters will be utilized in Tom King’s “Danger Street” series coming in May.
The American Way – Written by Academy Award winning screenwriter (and current I Am Batman writer) John Ridley, about an alternate history where the US government creates their own superheroes during the 1940s and how it changes things.
Blackhawk – Word around the campfire is that a movie based on the Blackhawks by Stephen Spielberg is in the works, so why not read some so that we can complain about how they changed things in the movie later? Especially when it’s from a comics legend like Howard Chaykin.
I don’t know how it would translate to a read-along, but what about picking a modern (or modern-ish) story that includes a bunch of obscure characters, and doing a read of all their original or near-original appearances? (I’m thinking of things like Lamplighter’s museum in Legends of Tomorrow or all the inmate common room in New Year’s Evil, etc…)
I’m gonna throw out another series that I think would work well here that I recently read: Camelot 3000. It’s historically notable for being DC’s first maxi-series and first foray into the newly developing direct market, as well as being the longest form series that comic legend Brian Bolland did in the United States market. Because it was for the direct market, it was also way more mature and willing to do things that just wasn’t done at the time, especially in terms of LGBT representation. And it’s just a damn good book.