I actually just did this a few days ago. While they are far from fleshed out, reading the originals of most of them leads to backgrounds that already exist (you'd need to pick appropriate bonds, flaws etc.). It's amazing how often noble seems to be the best fit (usually because they had the etiquette nonweapon proficiency and some sort of access to royalty)
(I also did a quick runthrough of PrCs from 3e; it was surprising to me how many of them were really just backgrounds tacked onto a mechanical foible-either fixing a tough mechanic or granting numerous small bonuses that really don't matter in 5e). After doing this little project, I realized even more that 5e really does cover everything quite well. (With the exception of animal companions, which show up a lot more than I remembered in 2e, or even 3e).
Backgrounds
These are from kits from 2e
Complete Book of Dwarves
Animal Master: Ranger
Axe for Hire: Soldier
Clansdwarf: Guild Artisan
Hearthguard: Soldier
Battlerager: Pirate with Intimidate instead of Perception (Barbarian)
Highborn: Noble
Alternate rule: if you spend an additional 25% on all mundane gear, you gain advantage on social checks with other nobles etc. Maybe a rule for everyone?
Outcast: Urchin or Criminal
Rapid Response Rider: Paladin or Ranger with mount.
Sharpshooter: Anyone with the sharpshooter proficiency
Craftspriest: Guild Artisan, knowledge cleric
Pariah: Charlatan or Criminal, trickery cleric
Patrician: see Highborn
Ritual Priest: Acolyte, knowledge cleric
Champion: Paladin
Temple Guard: Acolyte with Perception instead of Insight
Vindicator: Barbarian with Acolyte (with Intimidation instead of Insight)
Diplomat: Sage
Entertainer: Entertainer
Locksmith: As guild artisan but see below
Skill Proficiencies: Investigation, Sleight of Hand
Tool Proficiencies: Thieves’ tools, carpenter’s tools
Equipment: Thieves’ tools, as guild artisan
Pest Controller: As outlander, but see below
Skill Proficiencies: Animal handling, Survival
Tool Proficiencies: One musical instrument, Tinker’s tools
Ghetto Fighter: Urchin
Trader: Guild Merchant
Vermin Slayer: Ranger
Wayfinder: As outlander, but two languages and no instrument
Complete Book of Elves
Herbalist: Hermit, possibly ranger, possibly with Healer proficiency
Archer: Warrior with archery fighting style.
Wilderness Runner: Ranger
Windrider: Really a DM campaign choice. Maybe a ranger or paladin
Spellfilcher: Arcane trickster
Bladesinger: Eldritch Knight
War Wizard: Diviner or Evoker
Huntsman: Criminal with ranger
Collector: Sage (perhaps with a multiclassed AT and EK, or maybe lore bard)
Infiltrator: Charlatan (probably bard)
Undead Slayer: ?
Complete Book of Gnomes and Halflings
Breachgnome: Soldier with flaw 4, 6 or possibly “never surrender” (Battlemaster fighter)
Goblinsticker: “” (Champion)
Mouseburgler: Thief
Tumbler: Entertainer
Imagemaker: Illusionist with guild artisan (probably tinker gnome)
Vanisher: Illusionist
Buffoon: Noble (somewhat ironically), with entertainer bonds etc.
Stalker: Outlander Ranger with Archery style
Rocktender: Land Druid (mountain)
Treetender: Hermit land druid (forest) or nature cleric
Archer: Warrior with archery style, probably ranger, probably sharp shooter
Forrest Walker: Outlander, probably ranger
Homesteader: Hermit with Animal Handling instead of Religion
Mercenary: Soldier
Sheriff: Folk Hero
Squire: Noble Knight (but in reverse, with the squire doing everything), with Animal Handling instead of Persuasion
Tunnel Rat: Urchin
Bandit: Folk Hero with disguise kit instead of artisan’s tools
Bilker: Charlatan with a gaming set instead of forgery
Burglar: Thief (probably criminal, urchin or sage)
Smuggler: Guild Merchant with Deception instead of Persuasion
Urchin: Urchin
Cartographer: Sage, Outlander, Hermit
Trader: Guild Merchant
Traveler: Entertainer
Healer: Hermit
Leaftender: Druid (land), or nature cleric with outlander or hermit
Oracle: sage, either diviner or knowledge cleric
Complete Fighter’s Handbook
Amazon: Outlander (probably barbarian)
Barbarian: “”
Beast Rider: Paladin or Ranger with mount
Berserker: Pirate with Intimidate instead of Perception (Barbarian)
Cavalier: Paladin or Ranger with mount, noble (knight)
Gladiator: Gladiator
Myrmidon: Soldier
Noble Warrior: Noble
Peasant Hero: Folk Hero
Pirate/Outlaw: Pirate, Sailor, Folk Hero
Samurai: Soldier or noble
Savage: outlander barbarian, but could be arcane trickster totally flavored (not spells, but abilities)
Swashbuckler: Swashbuckler
Wilderness Warrior: Outlander
Complete Ranger’s Handbook
Beastmaster: Beastmaster ranger
Explorer: Outlander
Falconer: Ranger with falcon companion
Feralan: Totem barbarian outlander
Forest Runner: Folk Hero with disguise kit instead land vehicles
Giant Killer: ?
Greenwood Ranger (Limbant): Subclass?
Guardian: Hermit with survival instead of religion
Justifier: Spell-less ranger
Mountain Man: Hermit with survival instead of religion and artisan’s tools instead of a language
Pathfinder: Outlander
Sea Ranger: Sailor
Seeker: Sage
Stalker: Spy
Warden: Noble
Complete Paladin’s Handbook
Chevalier: Noble or Soldier
Divinate: Acolyte
Envoy: Noble
Equerry: Knight
Errant: Soldier or Folk Hero
Expatriate: Soldier
Ghosthunter: ?
Inquisitor: Mage Slayer feat
Medician: Hermit with healing proficiency
Militarist: Champion or Battlemaster with acolyte
Skyrider: ?
Squire: Knight in reverse
Votary: soldier or acolyte
Wyrmslayer: ?
Complete Wizard’s Handbook
Academician: Sage
Amazon Sorceress: Outlander
Anagakok: Outlander
Militant Wizard: Soldier
Mystic: Sage or hermit
Patrician: Noble
Peasant Wizard: Folk Hero
Savage Wizard: Outlander (possibly warlock)
Witch: Warlock
Wu Jen: Noble with extra flaws (taboos)
Complete Priest’s Handbook
Amazon Priestess: Outlander with Animal Handling instead of Athletics
Barbarian/Berserker Priest: Pirate with Intimidate instead of Perception
Fighting Monk: Monk
Nobleman Priest: Noble
Outlaw Priest: Folk Hero, criminal, pirate
Pacifist Priest: Acolyte, Hermit
Peasant Priest: Folk Hero
Prophet Priest: Sage, Hermit, Acolyte
Savage Priest: Outlander
Scholar Priest: Sage
Complete Druid’s Handbook
Adviser: Noble
Avenger: Fey Paladin
Animal Friend: Hermit with Animal Handling instead of Religion
Guardian: Hermit with Survival instead of Religion
Hivemaster: Standard Druid, shapechange into insects, instead of Woodland Stride, webwalker
Lost Druid: Hermit?
Natural Philosopher: Sage
Outlaw: Criminal
Pacifist: Acolyte, hermit
Savage: Outlander
Shapeshifter: Moon druid
Totemic Druid: Moon Druid
Village Druid: Folk Hero
Wanderer: Outlander
Complete Thief’s Handbook
Acrobat: Entertainer
Adventurer: Thief with criminal, charlatan, urchin
Assassin: Assassin
Bandit: Soldier, criminal
Beggar: Urchin
Bounty Hunter: Criminal
Buccaneer: Sailor, pirate
Burglar: Any
Cutpurse: Criminal
Fence: Guild Artisan or merchant
Investigator: Guild Artisan with disguise kit instead of artisan’s tools
Scout: Outlander thief or criminal ranger
Smuggler: Guild artisan or merchant
Spy: Spy
Swashbuckler: Swashbuckler
Swindler: Charlatan
Thug: Criminal Fighter or barbarian (or paladin?)
Troubleshooter:?
Complete Bard’s Handbook
True Bard: Lore Bard
Blade: Entertainer Valor Bard (with sharpshooting applying to daggers?); possibly battlemaster
Charlatan: Charlatan
Gallant: Knight
Gypsy Bard: Entertainer with Animal Handling instead of Acrobatics
Herald: Noble
Jester: Noble or entertainer
Jongleur: Entertainer
Loremaster: Sage
Maistersinger: Outlander with Animal Handling instead of athletics
Riddlemaster: Sage
Skald: Valor bard soldier or outlander
Thespian: Charlatan
Complete Ninja Handbook
Stealer-in: Monk or Thief
Shadow Warrior: Shadow Monk
Intruder: Spy
Consort: Charlatan
Pathfinder: Outlander
Lonewolf: Outlander
Spirit Warrior: Any Monk or maybe Arcane Trickster
Complete Book of Necromancers
Anatomist: Hermit (maybe switch religion for survival? Animal handling?)
Deathslayer: Outlander
Philosopher: Noble or acolyte
Undead Master: Necro Death priest