Some people who have purchased OS X 10.7 Lion are noting that they cannot get the downloaded installer to properly recognize their desired volumes for installing the OS. When the installer lists the available volumes on the system, it may say either, "This disk cannot be used to start up your computer," or, "Lion cannot be installed on this drive." This problem occurs even after people have run Disk Utility's drive-verification and permissions-fix routines.
It appears some instances of this problem happen with drives that either have been restored from Time Machine backups or have previously been used for Time Machine backups, so there may be a residual "Backups.backupdb" folder on the hard drive. This folder is used on Time Machine disks to store backups, and if for some reason it has been created or included on the target boot drive then the Lion installer might be recognizing the drive as a backup drive and not as one on which to install Lion.
Another fix people have used is to temporarily resize the drive's partition slightly. To do this, open Disk Utility and select the drive device. Then click the Partition tab, select the drive partition in the partition scheme diagram, and change its size by a few gigabytes. Save the changes by clicking Apply, and then revert them and click Apply again. With this complete, try installing Lion again.
Lastly, this problem seems to happen on computers on which people have installed Linux and other operating systems, so the drives may contain a number of hidden and custom partition setups that could interfere with the Lion installer. This may be as simple as not having an EFI partition on the drive (a hidden 200MB partition that contains boot information and options allowing the drive to be recognized by and interact with the system's EFI firmware).
To see all partitions on the drive, open the Terminal and type "diskutil list" followed by pressing Enter. This should output the drives by device ID and show the various partitions on them. If you do not see a partition entry that is named EFI, then you will need to partition the drive again to set up the EFI partition. To do this, back up your data (use Time Machine for the Mac OS, and use a cloning utility like Carbon Copy Cloner to save your alternative OS installations). Then boot to your Snow Leopard installation DVD, select your language, and open Disk Utility from the Utilities menu. From here, select the drive device and then both partition and format it by choosing "1 Partition" (or your desired number of partitions) from the partitions menu. Click the Options button to make sure the GUID partition scheme is used, and then click Apply to repartition the drive.
When the partitions are recreated, restore the Mac OS from the Time Machine backup and clone your other partitions back to their respective locations, and then try upgrading to Lion again.
Update (July 21, 11:25am PST): Apple has apparently been tracking these problems, and has issued a knowledgebase article covering the details and the partition resizing fix that we outlined here.