If the stock is publicly traded, there will be a record of it. If you bought it through a brokerage firm, they will have the record. The company should have a shareholders' relations department, which will be able to help you track the stock certificate and price for it. Otherwise, you might need to hit the local public library or university library and try to research it. They will have archived Wall Street Journals, etc. If you can remember the year you bought it, at the very least you can get the average price for that year.
The cost basis is simply the money you paid when you bought the security, including any commissions that you paid to acquire that security. If you accumulated stock over the course of many purchases, the total cost basis is still just the cost of all the purchases including commissions.
But be mindful of reinvested dividends. If a stock paid dividends and the dividends were reinvested, computation of the cost basis will require some work. All reinvested dividends need to be added to the cost basis, otherwise the cost basis will be much too low and you will pay too much tax. If the dividend payment and reinvestment records are not available, you need to reconstruct them. Find out from old Wall Street Journals or New York Times financial sections how much the dividend was each year since the stock was acquired or inherited, and use the number of shares and price per share on the dividend pay date.
A good web site to get free historical information on stocks is http://bigcharts.marketwatch.com/ but in order to use that you will need to know when you purchased the stock. You can then backtrack from what the value is currently.