1) I would not tease her about getting married.  Deep down in, I suspect that she does not appreciate that.  A good friend would be sensitive enough not to do that.

2) Perhaps she is feeling pressured when you try to set her up.  Maybe she would open up more if there were groups of people, married and single, which would include eligible men ?  I don't know - I'm just thinking ....

3) maybe she is someone who feels more comfortable writing  - maybe a suggestion to meet someone thru writing first - maybe that would be an easier ice-breaker for her.  Again, I don't know - I'm just thinking.....

4) I presume she attends a church (?).  Perhaps it is a small church with very few eligible men that she would connect with ?

I'm just thinking that encouraging her/ helping her to expand her social circles - without the pressure of dating - that would be the most beneficial to her.