Would like to make a friend of this newly-discovered relative

If you have her phone number I would give her a ring in a month or two and invite her to meet for coffee or lunch. It may take a couple of meet ups to see if there is a real connection on both sides or not. Nothing wrong with taking the initiative but you need to realise that you may be disappointed and she is just not interested in a friend/relationship.

I agree with Esspee, nothing ventured, however, as she is in her 80s it may be that she made a special effort to meet up on the one occasion and may not feel able to keep meeting.

The important thing is to let her know you are open to another meeting.
I would send her a pretty note saying how much you enjoyed meeting her and hearing about your family and that you feel so pleased to have been able to talk about your grandparents with her.
Then ask if she would be willing to meet up somewhere convenient to her for another chat.
You have nothing to lose.

I agree about the letter. I would leave it a little, then maybe find some photos, or little bit of information / family story and send it to her 'saw this and thought of you'
You don't know quite what her health or social situation is, she may not quite feel up to the kind of meetings you are thinking of. A little something like this isn't pushy, but leaves the door open.

I would write her a little note and say how nice to it to meet up and share family news. Do you have her address?

If you are able to write suggest you meet again in the near future. A letter is so much easier to do than a phone call as doesn't put people on the spot.

Do you have her address? Drop her a line saying how nice it was to be able to talk to someone who knew your family. Give her your details, tell her if she ever feels like getting together for another chat at some time to get in touch, then leave it up to her. Of course it depends on how mobile she is and how close she lives.

Can you phone and suggest another meeting?
If that happens, try to find things you have in common. If that goes well you could ask if she'd like to meet you occasionally.

Does she have your contact details? If so I would leave it up to her.

To me, this lady sounds pleased to have met you, then met up again and shared photos etc, but as far as she is concerned, that’s it, nothing much more in common.

Probably has no idea what you could keep chatting about if you subsequently met again, there’s a limit how many family reminisces you can have really.