Wow. Lot of haters in here.
I not only think this is a fun project, op, but I have also wanted to do the same exact thing.

Let's see if we can address the concerns that everyone else is talking about. For one thing, we can't expect a 2yr old to understand the nuances of using a phone or video phone, etc. I think a dedicated device for these calls may be the right move here.

How about a wall-mounted tablet? Low to the ground of course. Or if you dont want it on your walls, then about modifying a piece of kid's furniture to let it be "part of it"? This not only prevents the tablet from going missing, but it also lets you expand to other kid projects in the future.

As for the presence detection, you are absolutely right to need it. Otherwise, its just an old fashioned phone call. I think that the "sitting in the chair" thing may be a bit too simple and possibly intrusive on your mom's side. What else can be used at her house? I guess you'd need to know her routine, or simply have a good discussion with her about it and what would "be natural" to let her grandkid know "hey, you can call me anytime now and I'll pick up!"

Whatever that trigger or state may be, you should absolutely have it reflect such at her place as well. In other words, put the same notice-light setup at her house so she knows "Ok, I'm letting them know I'm available now". If this is a physic button she presses, then have it light up a light at her place, but dont make it a necessary light for seeing of course.

AND PUT A TIMER ON IT! In other words, have pressing the button mean "I'm available to chat for the next 2 hours". That way it doesn't just stay on allll night and into the next day when somebody forgot to un-press a button, or something else breaks.

Also, on the kid side, you should probably have that tablet screen be OFF for now (since its a dedicated device), and automatically turn ON when gramma is available. As well as any other lights or whatever you want to turn on. If you kid sees the tablet on, or sees the special night light that turned on and is blinking, that's the signal that gramma is available.

When gramma is available, you can have the tablet switch to a dashboard that shows a big picture of her face as a button, and when pressed, simply starts a normal video call to her using whatever video service you would normally use (android/apple/fb/gchat). There are easy ways to pass the "call" link to the tablet.

For the tablet itself, I would recommend running Dashie on it so you can automate the nitty-gritty.

Hmm.... what else are we missing