We have a mulberry tree that is getting ready to wreak havoc on our patio any suggestions for turning these lemons into lemonade?

This mulberry syrup looks good - with photo instructions. http://www.dirtykitchensecrets.com/sharab-el-toot-making-mulberry-syrup/

And maybe you could add some of this syrup to standard cocktails: a mulberry cosmopolitan, with vodka, soda and ice with a little lime, or a virgin cocktail with just seltzer and some slices of citrus.

We have a tree, too, though it won't bear fruit until well into the summer, when we literally have drunken raccoons lolling on the patio. Like prettyPeas, we have found that laying out a series of plastic-coated tablecloths under the tree and giving it a vigorous shake minimizes the disaster in our yard and also yields enough mulberries to really tackle some recipes. We add them to pancakes and muffins, make jams and sauce and compote and ice cream, and freeze any surplus berries. I'm surprised people seem lukewarm about them. We love them and find them as flavorful as other wild berries. I suspect you could use the sour peaches in recipes for sour cherries, gooseberries, or other sour fruits.

Luckily the mulberry trees at my parents house weren't dropping their purple rain on anything but the far edge of the property line, but I understand your situation. I found the best way to "pick" the mulberries is to get an old sheet/tarp and place under the tree and shake the branch directly over it. This should give you plenty, and hopefully prevent at least a couple from causing a drunk squirrel patio party.
I've made mulberry pie, mulberry-rhubarb crumble, some attempts at herbed mulberry jam, and simply cooked-down mulberry sauce from them. I guess if I were you I'd cook some down first, maybe add some sugar or maple, depending on your taste to see if you like the flavor enough for more involved preparations. Mulberries tend to be kind of bland and have a lot of seeds/stems (which you can remove with a strainer or food mill if you like). I enjoy them well enough to shake the tree at least once per season, but to me mulberries are not bad (which also means not great).

When life hands you lemons, make wine: http://winemaking.jackkeller.net/request135.asp

The mulberry tree at my last house was on the back corner of the property, so I didn't have to deal with the mess. While my boys and the chickens liked them just fine, I never found a wonderful recipe for mulberries and I was perfectly content to let them eat their fill. Jam, jelly, pie, muffins--boring. Mulberries, to me, don't pack a lot of flavor on there own, so they can't add flavor to anything you make out of them.

cut it down and use the wood in your BBQ. Seriously - we had one and unless you can colelct them all, it will only help get the squirels drunk (seriously, they wait for these berries to ferment and have a party).