Is either 星屑のロッド / 星くずのロッド [hoshikuzu no rod] ('stardust rod') or スターダストロッド [stardust rod] in Japanese. Note that the Japanese name is mentioned below only when it is the latter, as the first variation is vastly more common.
Final Fantasy IV
Stats: ATK 45, ACC 80, INT +10
Equip: Rydia
Buy: - (sell: 50,000 gil)
Treasure: Lunar Subterrane B4
Other: casts Comet when used in battle
Final Fantasy VI
Stats: ATK 180, MATK +7, STA +4
Equip: Strago
Obtain: ?
Description: Legendary rod tipped with a swirling orb of stardust
Other: Can be used with Two-handed
Final Fantasy IX

Stats: ATK 16, SPR +2, resist Dark +50%
Equip: Garnet (Dagger)
Abilities: Float, Reflect, Ability Up
Buy: 750 gil (sell: 375 gil)
Shop: Lindblum (Dragoos' shop (disc 2), Dragon's Gate merchant (disc 2)), Conde Petie (Thomas Goldpiler's shop), Black Mage Village (equipment shop (disc 3)), Alexandria (weapon shop (disc 3)), Treno (knight equipment shop (disc 3))
Steal: Zorn
Use: Madain's Ring
Description: Rod adorned with beautiful stars
Type: Rod
Final Fantasy XIV

Stats: DMG 74, INT +45, MCrit rate +35, MPOT +30, Enfeebling MPOT +20, Conserve MP +50, delay 4.3, DPS 17.21, blunt, Increases Flare DOT
Equip: level 50 (required), BLM
Buy: - (sell: 0 gil)
Obtain: A Relic Reborn (BLM path)
Type: Two-handed Thaumaturge's Arm, Unique, Untradeable, Convert: No, Meld: No
Other: スターダストロッド in Japanese
Final Fantasy Tactics
Stats: ATK 5, Parry 20% (when available), MATK +3, 25% chance of casting Gravity after successful hit
Range: horizontal 1, vertical up 2, vertical down 3
Equip (job): Black Mage, Summoner, Mystic, Onion Knight
Equip (character): Netherseer
Equip (enemy): -
Buy: - (sell: 5 gil)
Obtain: Rendezvous Mode multiplayer
Description: A rod tipped with a meteorite fragment.
Other: can be used with Dual Wield and Doublehand
Airborne Brigade

SSR
Type: Rod, Element: -, Growth: Slow, Cost: 31
Attack: 11,180 (lv1), 35,123 (lv200), ? (lv220), ? (lv240), ? (lv260), ? (lv280)
Defense: 11,450 (lv1), 35,973 (lv200), ? (lv220), ? (lv240), ? (lv260), ? (lv280)
Obtain: after Shattered World and Showdown, level Zidane ally to 99
All the Bravest
Stats: ATK +10, Number: 52, From: FF4
Equip: Black Mage, Magus, Time Mage, Summoner, Aerith Gainsborough, Shantotto
Drop: Mover
Description: The celestial glow of this rod bespeaks a power as unknowable as the heavens.
Pictlogica

Stats (lv1): STR +42, INT +32
Stats (lv20): STR +70, INT +52
Buy: - (sell: ? gil)
Obtain: -
Alchemy: Lilith Rod, Magicite Shard, Amethyst
Type: Rod, Rarity: ★★★★, Element: -
Record Keeper

Stardust Rod (IV)
Stats: ATK +35, MAG +84, MND +72, ACC 95 (lv1), ? (lv5), ? (lv10), ? (lv15), ? (lv20), ? (lv25), ATK +65, MAG +135, MND +122, ACC 95 (lv30)
Max Level: lv20 (initial), lv25 (first evolve), lv30 (second evolve)
Type: Rod, Rarity: ★★★★★
Other: enables use of the Dying Meteor Soul Break when equipped by Tellah

Stardust Rod (VI)
Stats: ATK +30, MAG +88, MND +44, ACC 95 (lv1), ? (lv5), ? (lv10), ? (lv15), ? (lv20), ? (lv25), ATK +60, MAG +139, MND +69, ACC 95 (lv30)
Max Level: lv20 (initial), lv25 (first evolve), lv30 (second evolve)
Type: Rod, Rarity: ★★★★★
Other: enables use of the Blood of the Eidolons Soul Break when equipped by Terra

Stardust Rod (IX)
Stats: ATK +7, MAG +23, MND +19, ACC 89 (lv1), ? (lv5), ? (lv10), ? (lv15), ATK +17, MAG +54, MND +35, ACC 89 (lv20)
Max Level: lv10 (initial), lv15 (first evolve), lv20 (second evolve)
Type: Rod, Rarity: ★★★

Stardust Rod (XIV)
Stats: ATK +30, , MAG +89, MND +44, ACC 95 (lv1), ? (lv5), ? (lv10), ? (lv15), ? (lv20), ? (lv25), ATK +60, MAG +140, MND +69, ACC 95 (lv30)
Max Level: lv20 (initial), lv25 (first evolve), lv30 (second evolve)
Type: Rod, Rarity: ★★★★★
Other: enables use of the Enochian Blizzaja Soul Break when equipped by Papalymo
Other: スターダストロッド in Japanese
Category: Equipment