How can I convince my family members to start by writing a diary in order to record (…)?
in order to record information about ourselves, historical context, etc (this having a genealogical purpose). As I am highly interested in my family history and there is a possibility that our future generations will be fascinated by their (our) family history as well.
diary (handwritten…), I hope that none understood I was making a reference to a blog..
This would also include dreams

I tried to do that for ages with my parents, but without much luck. It’s not that they didn’t want to do it… I think it’s just hard for most people to sit down and start randomly writing about themselves. I can write pages and pages of stuff about my ancestors, but when it comes recording things about my own life, I always get writers block!
What worked for me was giving my parents a "Grandparents Keepsake Journal" and asking them to fill it out for their grandkids. There are many of them on the market (see links below for samples) and not just for grandparents, but for anyone. They make it really easy to record memories, important events, family stories, etc. There are specific pages for them to write about their parents and grandparents; their childhood; how they met their spouse; special family traditions; tons of things. I think having those "prompts" makes it a lot easier to start writing! And the fact that there’s an "end" to it (you can fill out the whole book and be finished with your "assignment") makes it seem less overwhelming than writing an open-ended diary or journal.
Here are a few available at amazon.com, but you can probably find some at your local book store in the genealogy section:
http://www.amazon.com/Grandparents-Legacy-Your-Story-Words/dp/1404113312/ref=pd_sim_b_1
http://www.amazon.com/Grandparents-Book-Shared-Memories-Instruction/dp/076210984X/ref=sr_1_8?ie=UTF8&s=books&qid=1253226784&sr=8-8
http://www.amazon.com/Fathers-Legacy-Your-Story-Words/dp/1404113320/ref=pd_bxgy_b_text_c
http://www.amazon.com/Grandpa-Tell-Me-Your-Memories/dp/1563830388/ref=pd_sim_b_5