The reason INT 3 is used is that it's the only interrupt that has a single byte opcode (0xCC). Other interrupts require two bytes: CD <interupt number>.

This makes setting a breakpoint really easy, as all you have to do is replace a single byte (and restore a single byte) where you want to place your breakpoint. INT 3 being only one byte is also important when you're setting a breakpoint instead of a another single byte instruction - your newly set breakpoint won't override the consecutive instruction, which might be jumped to somewhere else in the code.