I would provide the relative with the same hardware and software that you use. Whether that's a phone, a tablet, a netbook, a laptop, or whatever - you will be their only technical support and "there should be a menu to let you adjust the volume, can you see anything that looks like a little speaker?" isn't going to work. You will need to be able to say "down in the bottom right, there is a triangle with a lot of little semi circles moving off from it to the right" or "on the Options menu, the 5th one down is Settings".
A visit that involves both setting it up and doing a test call from another room will take care of most of the issues. You can be sure that they have a microphone and a webcam and that everything is configured correctly.
Do not attempt to try to teach the person all the background, such as the difference between a web page and an app, or email and typing in the Skype chat area. Focus on the benefit - use a video call if you want to see the baby and for the baby to see you. Start with only one modality - video call or facebook photos or vine or whatever - and add more if the relative says "if only we could also ..."