The GHz of the board is not the factor you need to look at (depending on the logic board you are looking at).
You need to stick with a logic board that is compatible with your systems layout. Basically, this is a logic board within the series. As an example your MacBook Pro "Core 2 Duo" 2.4GHz 15" (Unibody) is MacBookPro5,1 so the 2.53 & 2.8GHz models within the MacBookPro5,1 series will work.
But before you go and swap out your logic board thinking that is why your system can't run a newer OS lets correct things so you can upgrade with what you have.
First, you do need to make sure your firmware is upto date. Follow this Apple TN to check and update: EFI and SMC firmware updates for Intel-based Macs.
Next you'll need to get an external HD to copy off your apps and data as we will need to reconfigure your HD as its not setup with the needed partition & format type. You will also need to install a bootable OS on this external HD. As it will be needed to run off of it to delete your internal drive.
You may want to think about upgrading your HD instead of reformatting it. This is a great time to do that! As it will make things smoother.
OK, What to get for a replacement HD? I would go with a SSHD vs a regular HD as it will give your system some zip! The best choice here is Seagate 2.5" SSHD. In any case the drive you do get needs to run at SATA II (3.0 Gb/s).
Then you can take your current HD and put it into an external HD case (ideally a FireWire one as USB is so slow) to use as your backup drive.
If you go this route first setup your new HD using the external case to house it first. Prep the HD and install via the OS installer App (Lion or newer). Then using the Keyboard short cuts by Pressing the Option key during startup and selecting the now configured new drive restart your system does it work? It should! Don't forget to alter the startup disk setting to your new drive before you shut your system down. Now you can go ahead and swap the drives over.
If you take the other approach (redoing your current drive) this is the point you can delete the partitions on your current drive and setup a new partition and format and install the OS you want to use.