Explain exactly where your work in your mentor's lab leaves off and where your own begins. Give full credit to your mentor's good work but explain that it has not gone far enough in a particular area where you are smart enough to strike a new path. Ask for the money you need to do the work, first being sure that you are not being overly ambitious. As suggested above for reeling in an overly ambitious project, figure out the costs for a typical experiment, figure out how many experiments you can do in a year, and multiply the numbers through. Don't try to do your work on a shoestring, but do not pad the budget either.
Reviewers are very good at detecting padding.
Talk to experienced investigators who know what is a reasonable total budget to request from the grant maker to whom you are applying, if this information is not explicitly stated. Swallow your pride and remember that asking your friends and colleagues for their criticisms makes a lot more sense than waiting to hear those criticisms from reviewers who don't know you and thus cannot have a personal interest in your future. Make sure you ask people who have the time to read your proposal and give them enough time to read your draft carefully, preferably ten days to two weeks for a 20-25 page research plan. This is difficult to answer until you have written a few proposals but rest assured that, like a home remodeling job, it will take at least twice as long as you expect. For a major research proposal such as an R01 grant for NIH, six months before the due date is not too early to start.