Thank you for your follow up response and for providing the Dell service tag, which is just as informing as knowing the motherboard. After entering the service tag at Dell's website, I found out that the system is a Dell OptiPlex 990. I researched further into the system and found out that the motherboard's chipset is Q67.
As the motherboard's chipset is Q67, unfortunately it only supports Sandy Bridge processors. Ivy Bridge support is only provided for Sandy Bridge motherboards with chipset H61, H67, P67 and Z68. Regardless of BIOS updates, the Dell OptiPlex will not support Ivy Bridge, it only supports Sandy Bridge as it's chipset is Q67. To conclude
your friend's motherboard will not support Ivy Bridge, which the i7-3770 is. Apologies to be the bearer of bad news.
The only way to support the i7-3770 you have would be to upgrade the motherboard too, preferably one with a form factor of uATX/microATX/mATX, since the case of the OptiPlex is quite small. The i5-2400 is still solid in performance anyhow, it only lacks Hyper Threading which the i7's posses.
Anyhow all the best.