A renegotiation of your rate, sometimes known as a float down, is a way of taking advantage of lower market rates that occur after you have locked your loan but prior to closing.  Each lender will have their own policy and some will not allow any renegotiations. 

If your lender does allow this option, you are normally charged 0.25-0.5% of your loan amount in order to take advantage of a lower rate.  This fee is typically added to your closing costs, but some lenders may require you to pay it out of pocket prior to closing.

Your lender is the best source of information regarding this subject.  You should have these discussions with your lender prior to entering into any rate lock agreement.

It depends on if you're working directly with a broker or a retail bank but, in general, there needs to be a marked improvement in the "cost" of your rate for the bank to negotiate. It costs lenders to lock rates for you, so they have costs -- paid by you directly or assumed by your broker/banker -- to lower the rate.

The rate must improve by at least 1/8%, and the cost is generally between .25% and .5% of your loan amount.

It's easy to learn about renegotiation. Call your broker/banker and ask what circumstances must happen in order to lower your rate and how costs are handled. If you can't renegotiate, ask how long it will be before you can refinance again. Good luck!