After a bit of searching and reading the documentation, it's clear that you can write user defined functions in SQL Server that are marked as either deterministic or nondeterministic depending on which built-infunctions are used within the body.
RAND() is listed under the nondeterministic functions (see msdn article). So why can't I use it in a function?
Using a View might work for you.
From Returning Random Numbers from a select statement
CREATE VIEW vRandNumber
AS
SELECT RAND() as RandNumber
The view is necessary because, as you already found out, a UDF cannot use the rand() function because that would make the function non-determistic. You can trick the UDF to accept a random number by using a View.
CREATE FUNCTION RandNumber()
RETURNS float
AS
BEGIN
RETURN (SELECT RandNumber FROM vRandNumber)
END
Finally, you can use this function in any SELECT to now return a random number between 0 and 1 per row:
SELECT dbo.RandNumber(), *
FROM Northwind..Customers