- Upgrade to Mavericks: it's free, and you can upgrade directly from 10.6.8 to 10.9. Then use Migration Assistant to restore the Time Machine backup. This is fully supported.
- Upgrade to Mountain Lion; if you don't already own it, get it here. Then use Migration Assistant to restore the Time Machine backup. This is fully supported.
- With some difficultly, you might be able to migrate your data manually. If you try this, you should probably clone your Time Machine backup volume first (just to be safe).
When I wrote this response, I assumed the old "Macbook 2008" supported OS X 10.8 and 10.9. Then I checked everymac.com:
- MacBook (2008 white or black): Maximum OS X: 10.7.5
OP did not specify which model the old 2008 MacBook is. If it's a unibody MacBook, or a MacBook Pro, or an "NVIDIA" MacBook Air, then the best option is to update to Mavericks (option 1), or if you really don't want Mavericks, then upgrade to Mountain Lion instead (option 2).
If it's a black or white MacBook, or an "original" MacBook Air, then it won't support Mavericks or Mountain Lion at all.