The title pretty much explains it. Any suggestions? I think she would prefer a movie but I want to show her korean dramas too... as for movie I don't know what to show her yet, as for dramas I can't decide which one should be first... I thought of I Year Your Voice; Kill Me Heal Me and I Need Romance 3... but I'm not sure? She likes romance and emotional. 
What genres does she typically like?
For movies have a look at: Pained (it made me cry and it was romantic), Miss Granny (not necessarily emotional or romantic but definitely a good movie for a first timer), Dancing Queen (it has romance and family love in, it made me cry of happiness) and Our Heaven (I cried. Plain and simple.)

For dramas I can only think of three right now which are: Super Daddy Yeol (about a mother, has romance and made me cry. What else do you need?), Descendants of the Sun (definitely romance and emotional in this one. Plus the editing and scenary is beautiful.)  and Scarlet Heart (not korean but it still was one of my favourites for quite a while, it has romance and very emotional, it is a historical palace setting though so except deaths and betrayal. Plus it's a bit longer then the normal drama. But it's very good.)

I can only think of these which I would recomend to some ones mother xD I probably could think of more but it's past midnight :)
For movies, maybe Finding Mr. Destiny or Love Forecast. They are both really cute romance movies.
The first drama I showed my mom was Goong and she loved it. My mom is not a tv person, but she watched that drama the whole way through with me and loved the OST. It was also the first drama I introduced to my sister. From that point on, my sister was a kdrama addict like me and my mom would always ask me from time to time what drama I was watching. I think Innocent Steps might be a good movie to start with. It's a cute romcom. 
My mum loves psychological stuff so i probably no hope what so ever haha

How about something like:

The masters sun or  The Greatest love.

Marriage contract was also very good and emotional. Easy to follow 
I had my mother watch The Greatest Love and she loved it! It was her first k-drama and she was a little reluctant but she really got into it.  She also likes romance so this might be a good one to introduce your mother to.
I would recommend Descendants of the Sun and Heirs.  Two great dramas that she should love and get hooked on if she loves romance and emotion :-)
I'd definitely recommend Another Oh Hae Young to my mom. I've only seen three episodes, though, but it definitely gives me the vibes of something she'd watch.