After creating your empire and selecting 'play,' you will be met with the 'Game Details' screen.


In this screenshot, I am guaranteed to receive the Cybrex.

Once there, select 'Advanced Settings' and scroll about halfway down until you see the 'Precursors' section.

By default, you probably have all of them selected. If you deselect all but one, you are guaranteed to receive it.

Note: I've heard reports of Precursors currently being bugged with the latest update. Further testing is needed to determine whether this is related to selecting just one or if it affects them in general. Either way, hopefully we can anticipate a patch.

• Relic: Vultaum Reality Perforator
  • Passive Effect:
    • −10% Pop amenities usage
  • Activated Effect:
    • Random combat modifier:
      • +40% Armor hit points for 2-5 years
        
      • +40% Shield hit points for 2-5 years
        
      • +40% Ship hull points for 2-5 years
        
      • +40% Ship weapons damage for 2-5 years
        
      • All the above for 1 year
• Home System:
  • Resource-rich system only
• Grand Archive Specimens:
  • Fossilized Vultaum Remains: +8% Consumer Goods from jobs if Individualist, +8% Energy from jobs if Gestalt
    
  • Vultaum Religious Pamphlet: +20% Governing Ethics Attraction
    
  • Vultaum VR Video Game: +7 Amenities
• Technologies:
  • Vultaum Reality Computer (unique building, capital planet only):
    • +1 Science Director
      
    • +1 Physics research from jobs (+2 with Archaeo-Engineers ascension perk)
• Permanent Empire Modifiers:
  • +7% Research speed (computing)
    
  • +10% Amenities
    
  • Additional modifiers if Fanatic Materialist (requires completion of additional special project; select option 2):
    • +10% Research speed
      
    • −20% Happiness

• Relic: Yuht Cryo Core
  • Passive Effects:
    • +100 Extra pops when establishing colony
      
    • +20% Energy weapon attack speed
  • Activated Effect:
    • -20% Ship upkeep for 10 years
• Home System:
  • Resource-rich system
    
  • Ruined Hyper Relay
• Grand Archive Specimens:
  • Yuht Audio Console: +20 physics research
    
  • Yuht Cryo Pod: +5% Habitability
    
  • Yuht Crystal Skull: +6% Trade Value if Individualist, +4% Energy from Jobs if Gestalt
• Technologies:
  • Atmospheric Filtering (not unique): +5% Habitability
    
  • Hostile Environment Adaptation (not unique): +5% Habitability
    
  • Yuht Astronomical Interferometer (unique starbase building):
    • +7 Sensor range (+10 with Archaeo-Engineers ascension perk)
      
    • +14 Hyperlane detection range (+20 with Archaeo-Engineers ascension perk)
• Permanent Empire Modifier:
  • Unlocks planetary decision 'Initiate Yuht Cleansing Process':
    • +10% Planet habitability

• Relic:
  • None
• Home System:
  • Relic World (size 25)
• Grand Archive Specimens:
  • First League Crew Manifest: +4% Alloys from Jobs
    
  • Khamdai Magil-Sha: +3 Stability
    
  • Underworld Cantina Sign: +20 Unity
• Technologies:
  • Weather Control Systems: +1 Housing from City Districts (not unique)
    
  • Anti-Gravity Engineering: +1 Housing from City Districts (not unique)
    
  • First League Filing Offices (unique building, one per empire):
    • +2 Administrator/Priest jobs
      
    • +50 Edict Fund (+100 with Archaeo-Engineers ascension perk)
      
    • Administrators/Priests produce +1 Unity and +1 Amenities
• Permanent Empire Modifiers:
  • +15% Research speed (industry)
    
  • +10% Planetary infrastructure build speed

• Relic: Javorian Pox Sample
  • Passive Effects:
    • +20 Years leader lifespan
      
    • Unlocks Javorian Pox orbital bombardment stance
      • 20% planet damage, 150% army damage, 150% pop kill chance
        
      • Does not affect mechanical/machine pops
        
      • Can create tomb worlds
    • Double chance to discover Biology techs
  • Activated Effect: +50% Biology research speed for 10 years
• Home System:
  • Resource-rich system only
• Grand Archive Specimens:
  • Irassian Bacterium Javoria: +20 Engineering Research
    
  • Irassian Kinder-Atlas: +3 Stability
    
  • Irassian Purge Order: +5% Leader Experience Gain
• Technologies:
  • Next researchable organic pops technology (society research if none available)
    
  • Irassian Naval Yards (unique starbase building):
    • +1 Shipyard capacity from all shipyards
      
    • +20% Ship build speed (+30% with Archaeo-Engineers ascension perk)
• Permanent Empire Modifiers:
  • +20% Research speed (biology)
    
  • +5% Pop growth speed

• Relic: Cybrex War Forge
  • Passive Effect:
    • Can recruit Cybrex Warform armies on capital (these are the strongest armies in the game).
  • Activated Effect:
    • Instantly convert 2k minerals into 1k alloys. This effect compounds with each use.
• Home System:
  • Ruined Ringworld
• Grand Archive Specimens:
  • Cybrex Mining Column: +6% Minerals from Jobs
    
  • Derelict Cybrex Warform: +10% Ship Hull Points
    
  • Kuurian Cybrex Blueprints: +5% Army Damage
• Technologies:
  • Cybrex Mining Hub (Starbase Building): +100% Mining station output (+150% with Archaeo-Engineers ascension perk
    
  • Mega-Engineering: Allows for the creation and repair of Megastructures (rare, but not unique)
• Permanent Empire Modifier:
  • +10% Megastructure build speed

• Relic: The Last Baol
  • Passive Effect:
    • +10% Pop growth speed if not machine empire, +10% Society research if machine empire
  • Activated Effect:
    • Unlocks Planetary Decision: Nu-Baol Life-Seeding
      • Terraforms target planet into Gaia World (does not require World Shaper ascension perk)
        
      • Spawns 400 Baol pops on target planet
        
      • Can only be used once; decision becomes locked again until relic is reactivated
• Home System:
  • Tomb World (size 20)
• Grand Archive Specimens:
  • None
• Technologies:
  • Epigenetic Triggers: allows uplifting pre-sapients (not unique)
    
  • Terrestrial Sculpting: allows terraforming uninhabited planets (not unique)
    
  • Baol Organic Plant (unique building):
    • +2 Farmer jobs
      
    • Increases farmer output:
      • +1 Food from farmer jobs
        
      • +0.5 Consumer goods from farmer jobs (+1 with Archaeo-Engineers ascension perk)
        
      • +1 Energy from farmer jobs (+2 with Archaeo-Engineers ascension perk)
  • Permanent Empire Modifiers:
    • −15% Terraforming cost

• Relic: Psionic Archive
  • Passive Effects:
    • −50% Breach the Shroud cooldown (psionic special project)
      
    • Enables edict: Zro Additives
      • +25% Leader experience gain
        
      • +30 Years leader lifespan if biological
    • Moderately increased chance of beneficial outcomes when breaching the Shroud
  • Activated Effect:
    • Choice of:
      • Increases ruler level by 1
        
      • Empire Modifier: +20% Ship weapons damage for 5 years
        
      • Empire Modifier: +5 Planet stability, −10% Edict upkeep, and +25% Government ethics attraction; for 5 years
• Home System:
  • Deposit of 5 Zro in system
• Grand Archive Specimens:
  • None
• Technologies:
  • Psionic Theory: −10% Empire Size from pops (rare, but not unique)
    
  • Zroni Storm Caster (unique starbase building):
    • +5% Shield hit points in system
      
    • 100% Shield nullification in system
      
    • −50% Sublight speed in system
      
    • −30% Accuracy in system
      
    • −30% Fire rate in system

• Permanent Empire Modifiers:
  • +40% Research speed (psionics)

• Relic: The Disturbance Oppressor
  • Passive Effect:
    • +15% Monthly physics research
  • Activated Effect:
    • Repels cosmic storms around empire capital (the size of the protected area scales with galaxy and empire size)
• Home System:
  • Desert world
• Grand Archive Specimens:
  • None
• Technologies:
  • Storm Manipulation (not unique):
    • Unlocks Storm Attraction/Repulsion buildings, edicts, and starbase buildings
  • Advanced Storm Manipulation (not unique):
    • Unlocks Advanced Storm Attraction/Repulsion buildings
      
    • Allows using a science ship to sustain Cosmic Storms
  • Hyperlane Detailer (unique starbase building):
    • +10% Sublight speed in system (+15% with Archaeo-Engineers ascension perk)
• Permanent Empire Modifier:
  • +10 Trade protection

• Relic: The Tempest Invocator (mutually exclusive with continental world home system reward)
  • Passive Effects:
    • +10% Monthly unity
      
    • −15% Ship upkeep
  • Activated Effects:
    • Create a Nexus Storm in a chosen system (range scales with galaxy and empire size)
      
    • +100% Cosmic storm repulsion effect for 10 years
• Home System:
  • Ruined habitat
    
  • Deposit of 3 Zro
    
  • Continental world with Patriotic Institute of Celebration (unique building) (mutually exclusive with The Tempest Invocator relic):
    • +10% Damage to shields
      
    • Each time a cosmic storm enters the system, 30% Chance for each pop on planet to gain the Storm Fused trait:
      • +10% Pop efficiency in a cosmic storm
        
      • +5% Pop happiness in a cosmic storm
• Grand Archive Specimens:
  • None
• Technologies:
  • None
• Permanent Empire Modifiers (optional):
  • +5% Monthly unity
    
  • 20% chance every 20 years that a pop dies and spawns a random cosmic storm (cannot be a Nexus Storm)

All of this information is from various pages of the wiki, compiled into a single one for ease of reference. If I missed anything, please let me know so that I may add it. Thank you.