Despite the description, the title is accurate. You must smelt
100 weapons, not enhance.
In order to smelt, you need to enhance a weapon to a multiple of 10, plus have a second copy of the weapon. Yoshitsune's Grinding
skill affects enhancement and smelting, giving more XP per weapon used as enhancing material and increasing the base success rate. Currently, Grinding can reach level 6 when Yoshitsune has an overall level of 20, and I would recommend reaching said level. Any numbers I give from here assume that you're there.
The best weapons to use are common weapons found in early dungeons in Onigashima. Basic level 1 gear is Sukedachi Sword, Nora Twin Blades, Stone Battleaxe, Cedar Spear, Bamboo Bow, Pine Staff, and Manjusri Talisman. This will be significantly cheapest to smelt. Note that there are uncommon variants that you can get as well, Flame Sukedachis and Manjusris of Flame, for instance, that will cost significantly more.
You will also need rare weapons with gold icons. These provide the most XP, and you will need four gold weapons to max out a weapon to +10. There are three options here. The first two, Kikuichimonjis and Kikusenkakus, are found in the same early weapon drops. The third option is Oninodachis, which you'll likely end up crafting several hundred anyway.
While the ingredients can be found in various areas around Onigashima, the big grind here will be Onigashima Training Grounds
on Hell. This gives both the Sinister Bones needed for crafting the Oninodachi as well as increased weapon drops. The second chest in the training grounds always, or close to always, has a weapon inside, sometimes multiple weapons.
So you've got a good stash of common weapons as well as gold weapons. Time to smelt! The best way to do this is to constantly smelt to +10 rather than smelting the same weapon repeatedly. To conserve supplies, use your previously smelted weapon as materials. At level 20, the base smelt success rate is 90%, and if you want, you can opt to pay to reach the maximum 95%. The total will be... 237 ryou. Uncommon weapons start around 6k and gold weapons around 10k, so again, you want to use these as enhancing materials instead.