I've installed a new Seagate 1TB SATA hard drive in my 7 year old computer which has 2 EIDE ports using a SATA to IDE adapter into the secondary port, but the BIOS does not recognize the new drive. Any thoughts on what I might do to help this install along?

7 years?? That's pretty old! Anyway, I don't get why it wouldn't be seen in the BIOS. Other than buying a new computer, I suggest that you choose a different connection method

Swap the mainboard, or at least look for a BIOS update on the current mainboard manf. Website. If the BIOS can't handle the size of the drive, it won't report it - this is the most likely scenario.