Not a matter of memory, just how the USB ports are powered (or not) by design. That HDD just needs more juice than those ports can offer. On other systems the ports that work are the ones that can deliver. Not unusual to have it work only on one port but not the others on the same system. If there are other peripherals that hook up to the laptop at the same time, it could be that by design these ports are actually linked in parallel, meaning acting like a USB hub. They all would connect to the same USB motherboard connector. In that case, even if it seems that you are connecting different peripherals in different USB ports, it actually connects those peripherals to the same port, as they all have to use the same power supplied by that connector on the mobo.
I don't know if that is a clear way to explain it, but i tried.
anyhow, do you have other peripherals hooked up to the USB ports at the time you are trying to connect this HDD? If you do, try disconnecting them all and then connect the HDD. Try all the available ports on the laptop. If still no luck, that's the laptop's design and there isn't much you can do, except maybe get an external HDD that has it's own power supply.
There are HDD's that require that kind of power from USB in order to run properly. I have one like that and it is a PITA to have it running on my laptop. I have one USB port on one side and 3 on the other, but it turns out that the 3 of them are hooked to the same mobo connector, and they can not run my HDD. I have to connect to the single one to make it work - even so, it sometimes randomly disconnects and reconnects, guess it's pushing my USB port to it's limits in terms of power delivery.
Hope all this helps.