Let's say your list  is in column  A, starting in row 10.
I'd use:
StartRow = 10
RowCnt = Application.WorksheetFunction.CountA(Range("A" & StartRow & ":A65000"))

This will count the number of non-blank cells  in column A from row 10 to 65000
Keep in mind that if there are any blank rows in the middle of the data, it will
NOT
count it, and the number will be off.
Next use:

Randomize 'Initializes the
RandRow = Int((RowCnt + 1) * Rnd) + StartRow

this should give  you the Row Number of a random  row in your range.

Can you put it together from here?

I've got the code  working fairly well, but for
some reason it sometimes highlights the blank cell  after the last cell
containing data. For example, StartRow is 2, and cells  A2:A5 have data in them,
but sometimes cell A6 is highlighted. Any idea what I've done wrong? My code
is as follows:

****************************************************************
Sub HighlightRandomCell()
Columns(1).Interior.ColorIndex = 0
StartRow = 2RowCnt = Application.WorksheetFunction.CountA(Range("A" & StartRow &
":A65000"))
randrow = Int((RowCnt + 1) * Rnd) + StartRow
Cells(randrow, 1).SelectCells(randrow, 1).Interior.ColorIndex = 6
End Sub
****************************************************************

I suspect it's the "RowCnt + 1".... try using Int(Rowcnt * Rnd) and see how it
works.