Are you are trying to reinstall using a restore which downloads from Apple?
Instead reinstall directly off of the CD or DVD set that came with the machine.
Then if you are at revision 10.6.8 or higher
you can download the El Capitan update installer and upgrade to 10.11.6.
How to Upgrade to OS X El Capitan - Apple Support
https://support.apple.com/en-us/HT206886
Download Link to OS X El Capitan
https://itunes.apple.com/app/os-x-el-capitan/id1147835434?ls=1&mt=12
Read
Update Information for El Capitan - a tip series by .A Brody
https://discussions.apple.com/docs/DOC-8811
*** n.b. The install deletes the install file after it is done so make a backup copy first!
And - MOST IMPORTANT - of course backup your system first !!!
If your machine does not meet the requirements for 10.11.6 or
for some reason you do not want to upgrade that far
Then try to upgrade the OS from - The Apple - About This Mac - Software Update
An upgrade might work better than a direct install of Lion.
I would suggest upgrading all the way to at least Mt Lion (10.8.5) if possible.
There is no way to upgrade to 10.9 or 10.10 unless you have a save file from doing that before.
It is up to 10.8.5, or next step to 10.11.6, or up to the MacOS versions.