The Weapon Connoisseur
Nebb is available running around the middle of town roughly between noon and 7 p.m. After starting the quest, he can be located by setting The Weapon Connoisseur as the active quest in the Adventure Log.
Talk to Nebb to start the quest.
Tip: If you have done the Sunshroom Sensing Side Quest, have Sheikah Sensor +, and have taken a picture of the requested weapon, you can set the weapon as your Sheikah Sensor + target for help in finding one. If you have not taken a picture of the weapon, you can talk to Symin at the Hateno Ancient Tech Lab and purchase equipment pictures for your compendium (at 100 rupees each, getting a random picture each time, can buy in bulk if desired) until you have a picture for your next objective. You can then set it as your Sheikah Sensor + target for help in finding one.
Each of these weapons likely have multiple locations. The locations listed below are simply suggestions.
If Link is already carrying the requested weapon, speak to Nebb again to show him the weapon. It does not need to be equipped.
Objective 1: Traveler's Sword
A Traveler's Sword can be found on the path just east of Inogo Bridge. (If using Sheikah Sensor +, this is #204 in the base game, #207 with the Breath of the Wild Expansion Pass DLC and #212 on Master Mode).
Show it to Nebb for a red rupee.
Objective 2: Fire Rod
Show it to Nebb for another red rupee.
Objective 3: Moblin Club
Show one to Nebb for a purple rupee.
Objective 4: Duplex Bow
Show one to Nebb for another purple rupee.
Objective 5: Windcleaver
Windcleavers can bound on Yiga Blademasters. They are available in various places (Hyrule Field, Akkala Highlands in particular) but become more available after clearing the Yiga Clan Hideout. (If using Sheikah Sensor +, this is #279 [base], #283 [DLC], #288 [Master Mode] in the Compendium.)
Show it to Nebb for a Silver Rupee.
Objective 6: Ancient Battle Axe+
The Ancient Battle Axe+ drops off Guardian Scout III, which can be found at many of the "Modest Test of Strength" shrines. As long as a blood moon has passed, the Guardian Scout will respawn, even if the shrine has been beaten. (If using Sheikah Sensor +, this is #274 [base], #278 [DLC], #283 [Master Mode] in the Compendium.)
Show it to Nebb for another silver rupee.
Objective 7: Frostspear
There is a Frostspear in a chest at the shrine in the Spring of Wisdom (Jitan Sa'mi Shrine). If that has been collected; a Stalnox in the Lake Tower region has one, or one of the Lizalfos in the Coliseum Ruins once the World Level is high enough. (If using Sheikah Sensor +, this is #302 [base], #306 [DLC], #311 [Master Mode] in the Compendium.)
Show it to Nebb for a Gold Rupee.
Objective 8: Ancient Short Sword
Show it to Nebb for a diamond, and you will finally complete this quest.