inherit time machine backup after hard drive replacement?

My daughter's 2009 MacBook hard drive failed. Luckily Time Machine was on the scene and after replacing the drive, we were able to restore her user account via Migration Assistant. But we weren't given the option to inherit the Time Machine backup, maybe because we weren't doing a full backup (data only). Is it possible to "reconnect" her machine to the old backup? If no, can I simply delete the old backup files and start a new one?

The MacBook is running Mountain Lion and shares a wifi connection via Airport Extreme to a Seagate 1.5TB FreeAgent Desk external drive. The drive has seven partitions, one of which we're using for her MacBook's Time Machine backup. The other partitions are used by other machines.

Thanks for your help.

Have you tried backing up to Time Machine selecting the previous MacBook partition that you've been using before the HDD replacement? If so, what sort of message are you receiving?

Also what version of the operating system are you using? The early version of TIme Machine that shipped as part of 10.5 were sometimes difficult to reconnect with an existing Time Machine backup. Later versions seem to have fixed the problem.

The MacBook is running Mountain Lion, OS X 10.8.2. I tried selecting the former TM partition, but the results are a little confusing.

First, the backup failed with this message: "Time Machine error - this backup is too large for the backup disk. The backup requires 130GB but only 30.57 GB are available." But in Finder, the drive shows as "7.81 GB of 129 GB available." I originally sized the partition at 130GB.

Before the backup failed, I noticed the message "deleting old backups." The oldest backup used to be from April 2011, but now the oldest backup and latest backup have the same date from December 2012. So it seems to have found the backup file and tried to use it, but apparently didn't have enough space even after clearing all but the most recent backup.

When I originally configured Time Machine, I wanted to backup user files only, so I excluded Applications, Library, and System Files. But 10.8 doesn't appear to offer the ability to exclude System Files. Is that why the backup is so big? Can I exclude System Files, or am I doomed to partition the drive all over again?

Thanks again for your help.