When you configure the BGP Dampening, the configuration is set for all eBGP neighbor in a Cisco router. There is a way to customize the parameters for a specific neighbor ?
You can set bgp dampening for an AS path, which accomplishes the same thing. For example, if your neighbor is AS 1234, you can do something like this:
ip as-path access-list 1 permit ^1234 route-map damp_me permit 10 match as-path 1 router bgp 65000 bgp dampening route-map damp_me
Important Note (edited):
Route dampening recommendations have gone through several changes since first created. At one point, it was recommended not to use it, as it caused more problems than it solved.
Now, the latest IETF BCP (2015) recommends to enable RFD with adjusted values (right-most column), which are different from the vendor defaults.
RFD parameter Cisco Juniper BCP 194 / RIPE-580 Withdrawal penalty 1000 1000 1000 Re-advertisement penalty 0 1000 0/1000 Attributes change penalty 500 500 500 Suppress-threshold 2000 3000 6000 Half-life (min) 15 15 15 Reuse-threshold 750 750 750 Max suppress time (min) 60 60 60
The table is copied from this article which also reports about current deployment and RFD's history.