If you have travelled back on the exact same timeline, the most effective way to convince your family of your validity is to share childhood memories. If you, the time-traveller, have any physical scars (from surgery, childhood accidents etc) those will assist in convincing your family that you are not a mad man/woman... but sharing memories from your past is also age dependant, as the memory gradually diminishes with age. If, you, the time-traveller is in their 70's, then it may be just that little more difficult to recollect, childhood memories. But whatever age you travel back, you not only have a lifetime of memories to sift through, you will also be actually reliving those memories, which I imagine would have a psychological impact on the mind. Imagine living a life, where you will most likely know what is going to occur next (dependant on memory of course)... I suppose it could be like having successive feelings ofDéjà vu (already seen).
Then there is also a paradox to consider. If, in your hypothetical scenario, the time-traveller (Mr X.) travels back to visit his family, which includes Mr X as a child (little x), then Mr X should already have the memory of that encounter. Logically speaking, little x, while growing up, will carry memories about an adult (Mr X) trying to convince him that he was the future little x.
This is slightly off topic, but because I'm interested in religion/philosophy etc, I need to ask this....Assuming that the spirit/soul does exist, if Mr X possesses 1 spirit/ life force, how could he revisit himself in the past? That meeting would involve Mr X talking with a younger version of himself, (little x). Mr X is one person, but now there are 2 human organisms/bodies of the 1 same person. There are 2 souls/life forces present of the 1 person.... how can this be?