Two ways come to mind.

1) Use the "Modified" date  field. Of course this only works if nothing
else will be modified after the box  is checked.

2) Build a small custom workflow with SharePoint Designer. You'd need a
separate Date/Time field  (*not* calculated). Tell the workflow to run when
an item is added or changed. In the Conditions section of the first step --
really, the only step -- check  to make sure that (a) the field value is Yes,
and (b) the DischargeDate field is empty. If those two are true, then
populate the DischargeDate field with the current date.

Although you can build a calculated column  that will store the date  if the
checkbox is checked  you wouldn't be able to tell it not to update the
calculated column if the checkbox  starts out checked when someone re-edits the
item. You also can't use a workflow since workflows only fire after an item has
been saved. Again, you can tell in the workflow whether the checkbox is now
checked, but have no way to know if it started out that way. The only way to do
this would be by writing an event Receiver that would fire while the change is
being committed to the database. Then you can determine if the state of the
checkbox has changed from unchecked to checked and set the date field
appropriately. I'm afraid in this case custom programming is the only solution.

Create a workflow on changed.
Check if box  is checked  and date  is empty else stop
Update the date box with the modified date.

The date will need to be a date field  not a calculated  field. Can be either
date or date/time.

I guess my messages are coming through slowly...?