The main ways are the same as for other types of subjects:
- Be stronger than they are (pump your chest by building over force limit if needed)
- Opinion sitting at 200 (improve relationship)
- High dip rep (Influence, Diplomacy, trading in Ivory bonus, policies)
- Trust sitting at 100 (use the Great Power "Influence Nation")
- Develop subjects from time to time if needed (temporary -5% liberty desire)
- Placate subjects from time to time if needed (temporary -10% liberty desire)
- Grant them provinces through the subject interaction screen (mostly for other types of subjects, but this should apply to colonial nations too if you're paused on the day you make peace; temporary -% liberty desire based on the province's value)
- Pay off their debt whenever possible (temporary -% liberty desire based on the number of loans)
If you're filthy rich you can abuse the latter by loading large subjects up with forts, to make it so that they're constantly running a deficit. And then pay off their debt every so often. Be sure to stay constantly at war if you do this, else they'll delete excess forts when at peace. (Pause to stay at peace for a single day if you ever need to do something that requires you to be at peace.)
As OPM Hamburg you can keep Eastern America as a happy colonial nation if you apply the above.