SUM(DISTINCT) Based on Other Columns

select sum (rate)
from yourTable
group by first_name, last_name

Edit

If you want to get all sum of those little "sums", you will get a sum of all table..

Select sum(rate) from YourTable

but, if for some reason are differents (if you use a where, for example) and you need a sum for that select above, just do.

select sum(SumGrouped) from 
(    select sum (rate) as 'SumGrouped'
    from yourTable
    group by first_name, last_name) T1

David said he found his answer as such:

SELECT SUM(rate) FROM (SELECT * FROM records GROUP BY last_name, first_name) T1

But when you do the GROUP BY in the inner query, I think you have to use aggregate functions in your SELECT. So, I think the answer is more like:

SELECT SUM(rate) FROM (SELECT MAX(rate) AS rate FROM records GROUP BY last_name, first_name) T1

I picked MAX() to pick only one "rate" for a "last_name, first_name" combination but MIN() should work the same, assuming that the "last_name, first_name" always leads us to the same "rate" even when it happens multiple times in the table. This seems to be David's original assumption - that for a unique name we want to grab the rate only once because we know it will be the same.

SELECT SUM(rate)
FROM [TABLE] 
GROUP BY first_name, last_name;