Science lesson everyone!
Electric motors work on the reverse principle of an electric generator. Basically an electric generator requires you to move a magnet around some wire. The faster you move the magnet, the more current you generate. In an electric motor, a current passes through some wire and creates a magnetic field, which opposes against some magnets, creating motion. The more current you have, the stronger the magnetic field and thus the stronger the opposition against the magnets inside.
So what does this have to do with an LED blinking? It turns out that electric motors create draws pulses of current. This is just part of the electric motor design. A smart electrical engineer would have the LEDs and motor wired parallel. A cheap electrical engineer would just wire them in series. Since there are times when there's no current drawn (because the electric motor isn't drawing current), the LED can't be turned on.
So yeah, unless you know what you're doing, it's probably better to just get another fan that doesn't have this problem.
This could very well be the case, or the OP could have them attached to a Pulse Width Modulation controller, which works by cycling the electricity on and off really quick. This can also cause LEDs to blink or flash.