How can I gain silver coins quickly?

There are two easy options.

• dig deep, mobs spawn like crazy down there, and they are harder then the mobs on the surface = better loot
• find the corruption and park yourself on the edge of it, you will be attacked by soul eaters, super easy to kill and they drop silver coins most of the time

Although not the fastest way to "farm" silver coins, as such - could you just invite a friend who already has 50 silver coins into your world as a multi-player environment, and then the merchant will spawn assuming you have the right house for him? A world made available on multi-player will still be available for you as a single player afterwards.

As Camster points out below, they would have to have 100 coins, and give 50 to you - as each player needs to have 50 coins before he will spawn.

Since you won't have access to a merchant, the best way would be to get some decent equipment (i.e. a copper sword, though wood would also be fine) and go spelunking. Get as many pots as you can.

I personally just got lucky and found a pinky slime on my first day - 1 gold coin.