Talk about old times:
Most elder relatives crave to talk their heart out to some one because most of them live a retired life and feel aloof when every one else around them is working. They want an ear to listen to their feelings and maybe even complaints. Many will want someone to discuss their health issues with. Give them this luxury and make them feel good by letting them clear their heart and comforting them. Many relatives would want to discuss how times have changed and how times were better when they were young. Hear them out. Some might be excellent historians and narrate you stories from their time which will increase your knowledge about the past and glean information about other family members.
Go through family albums:
For many elders out there, family albums are the best source of entertainment. They see them as a record of their life and there are memories attached to each photo. Go through these with them and stop at each photo to talk about what events went behind it, any trip, an occasion or when you all randomly chose to take photos. This way they will also realize how loved and lucky they have been through the years.
Play board games:
Board games can be quite interesting pastime. They do not usually require any physical activity so any one can participate including the elderly relative. They might just have the edge due to their experience.
Tell them stories from your life:
Fill them up with stories of your life. Your personal life, work, friends and what goes around the family. Update them on everything. Elderly relatives usually adore the kids in the family. Tell them stories the teachers have told you from school. The kids do not necessarily have to yours. Any kid that the elder is fond of will do. Go back in time together when you and your young siblings and cousins used to do the same.
Read out chapters from their favorite book:
If the relative has developed eye sight problems or cannot read any longer, you can read out chapters from his/her favorite books. These may be poems or any other content that they like. Some even might wish to be read out the Bible, the Quran or any other Holy scripture. For them it might be the most satisfying experience.
Cook and play for them:
If you are good in cooking, ask them their favorite dish. Place a chair for them in the kitchen and start cooking. Play their favorite music in the background and chat away with them while working. Eat the meal together. This can be the best form of entertainment and make some more memories. You can also sing their favorite song on the guitar or yourself if you are good at music.
Help them re connect:
Ask them the people that they miss. They might be old friends and far away relatives. Trace these people using the internet or mail. Write emails and letters to them from your relative's side inviting to re connect and cherish old memories. You could help in just tracing the address and providing the relative with it if he/she is able to mail him/herself. However, if you know a friend they really want to talk to, surprise them by putting a call from your mobile and holding it to their ear without telling them beforehand.
Take them to walks:
If they are unable to walk, help them out and take them to the park and entertainment places. Even to the cinema. If they are able to walk themselves, just give the company and take them to their favorite places in the city, a cinema, restaurant or to visit a friend.