Leave it standing -
If it's not disturbing you in any way, and it's stable, then there is nothing wrong with leaving the tree standing.
Free firewood -
If the tree is dead anyway, you might as well use the wood to keep you warm in the winter.
Sell it -
Contact a local company and sell the lumber. Depending on the type of tree, you might even make some good money!
Hanging bird feeder -
Use your dead tree to feed the local birds. Hang a bird feeder and see them returning every day to your garden.
Tree stumps make great seats. If the tree is thick enough, you can also turn it into a coffee table. You can sand and varnish them for a nice finish.
Cut a hole in the stump, fill it up with soil, and voila!
A wooden bench can be as elaborate or as simple as you want it to be.
You can cut your dead tree and use the wood to build garden beds or walkways.
Stepping "stones" -
Simply slice up the tree trunk and use the wood as garden stepping stones.
Use the old wood to create a stamp set for your kids to play with.
Bird-nesting box -
Give the birds around your house a nice cozy space. It's always nice to have wildlife in your garden.
You can hang a birdbath on the tree or even carve one if you want to.
Floor lamp -
You can use branches to build a lamp. Let your creativity (and electricity) flow!
Wall clock -
You can easy build one, even if only for decor purposes.