If the LaCie unit is eSATA, then the requirements for your new desktop would simply be "eSATA". It's an external port on the back I/O panel, just like a USB port would be. The only difference is that eSATA is relatively new and has been difficult to get going. It is not a common port right now by any means! However, if your intention is to get a new desktop system to go with your new RAID 1 external enclosure, then matching them up should be easy! Just look for a computer that boasts eSATA. I'm sure Dell and others have plenty such systems available right now, even if they do cost a bit more than the "cheapie" computers.
In terms of preferred RAID capabilities, why is it that you want an external RAID 1 enclosure? Capacity? I ask because RAID 1 is fairly common these days internally, although if you don't know a lot about computers then an external enclosure is pretty fool-proof. It's also portable, I guess, if you take precautions, but I don't know if this is important to you?
On a side note, don't let anyone push you over as far as internal SATA being faster than eSATA! The 300 MB/s bandwidth limit is not going to be exceeded by any external enclosure nor internal hard drive! Modern hard drives push a mere 55-65 MB/s continuous on a good day! 5 of them through a single SATA or eSATA port would be acceptable, so don't listen to anyone who tells you that 1 drive vs 2 is going to make a difference!