The questions above are important, as is the dollar amount of the inheritance for comparison purposes. It is relative in terms of earning potential and debts. What is the inheritance amount in terms of monthly family income? One month's worth or one year's worth makes a big difference, for example. Would the inheritance pay off any debts?
Ultimately, I'd split the inheritance between debt and savings...versus all of it in one or the other. The percentage or dollar amount to each would depend upon your family's bigger financial picture. The most "lopsided" I would go is an 80/20 split in either direction. I'd lean more towards 60/40, though, in either direction (specifics dependent).
Also, check into the prepayment terms of your mortgage. Some lenders have strict policies and others are open. You might incur a penalty for paying a lump sum towards the mortgage, but could avoid that by paying up to "x" amount extra each month towards principal (until the inheritance amount is used).
In addition, there are some folks who are concerned about how their children's college funds are set up. All money in the child's name is expected to go towards the child's education, for example, when applying for financial aid last I checked. Money in the parents' name(s) is only counted "against" the child at certain percentages. Tax-sheltered investments can alter the effects to a certain degree.
IMO, fear isn't a good place to begin making decisions. It is good to recognize it (fear) and approach it as healthy and an indicator for "need more information". One's relationship with money is more important than how much one has or doesn't have, in my experience (which is years of education and working with people and their money).
One wise course of action is to deposit the money into a interest-earning, insured account until you can do the proper research. If you have a Costco membership, they offer bonuses for two accounts via Capital One banking and one Sharebuilder investment account. ING Direct offers an online savings account for a decent (for this financial climate) interest rate. If you search online for high rates on online savings accounts, be sure to research the "bank" before sending any money. The ones I mentioned are legitimate and have solid reputations. There are other good ones, too. And, unfortunately, there are plenty of non-legitimate places, as well.