No, it will not melt. Brown sugar has an even higher decomposition point then white sugar when making caramel, and it does not dissolve in water easily at all. Think about the last time you made chocolate chip cookies. You know that point when it's just the butter and sugar? You know, the magic delicious time? It's still gritty because brown sugar does not break down until it's been blasted in high heat. I've used graham crackers, brown sugar, and all kinds of other bits on my buttercream cakes and they are just fine.
Note: Dark brown sugar is the less refined and contains has the most molasses is it, so it has a higher decomposition point then light brown sugar, which is 1/2 way refined white sugar. I generally use dark brown for any decorations. Actually, for sand I do 1/2 dark brown sugar and 1/2 graham crackers so it's not just clumps of sugar of the cake. For added texture chuck in a bit of sugar in the raw or some clear sanding sugar.