This is one of the 2 places I have found at the start of the game where the weapon you will find is randomized. Both places are in the Cave of Fal. The other point isn't in a room. Its the waterfall before you have the scorpion boss fight. Walk into water beside rabbit and click. Weapon is random there, best one I have found so far was for Guandia. Among other weapons randomly acquired here was a Crystal Claw Pwr 58 Holy Excellent vs Humans and Dragons. The Sword in the mirror is randomized as well in room 6. Depending on individual patience in repeating each of them, I have no idea of how many potential weapon variations are available. The one beside the rabbit can potentially be that junk Crude Sword for example.
Swords I have gotten in room 6 are as follows:
1: Water Sword Pw 137 water /may poison
2: Air Sword Pw 139 air /may cause bleeding
3: Thunder Sword Pw 141 thunder /may Numb
4: Ice Sword Pw 148 ice /may cause cold
5: Earth Sword Pw 152 earth /may blind
There might be other possible swords there, but I figured that is a good enough list for now anyway.
Doc
I edited to give the list of swords.