Someone who's suggestible has an opinion that's easily swayed. If you mention to your suggestible uncle that it might be fun to get ice cream, he'll probably drive you to the ice cream shop.
It's easy to convince a suggestible person to do something, believe something, or to change their mind about something. Suggestible voters are easily swayed by emotional political ads on TV, and if you're very suggestible, your brother talking about his migraine can convince you that you might have migraines too.