Probably the best way to accomplish this is by purchasing stock directly through the company you're interested in by their "dividend reinvestment plan". "DRIP" investing is a very low cost way to purchase small numbers of shares in a particular company, and then automatically reinvest the company's dividends in additional shares.

Say you want to buy stock in a family movie/amusement park company that your little niece or nephew just loooovvve. Go to the company's website and look for the "Investor Relations" tab. You'll find the minimum purchase required (for a lot of companies it's around $250) and all the paperwork. Of course, with today's stringent Patriot Act security, you're not going to be able to surprise your recipient with the stock -- they'll have to actually open the account. But they can get a really cool stock certificate (or hold it directly with the transfer agent for safe-keeping) and the ability to add to their investment over time.

Thousands of companies participate in these direct stock purchase programs.

You must really like your family. I don't know about transferring stock, but one way to do it is a donor-advised funds.

Check out - http://www.filife.com/stories/making-giving-part-of-your-financial-plan

"Because donor-advised funds are easy to run and can be started with small amounts of money, a growing number of affluent parents and grandparents are establishing them for their children and grandchildren as a way to teach them about giving. Others let the money in their own funds grow tax-free until children and grandchildren are old enough to help choose worthy charities."

Thanks Hal! Great answer and I always wondered how people end up with a paper stock. I just sort of figured that with the days of electronic trading paper stocks were no longer issued.