Giving percentage of my LLC but not the income generated by the rental.

There are many ways that you could accomplish your goals. However, aside from the tax perspective and need for advice from that angle, you should contact a wills, probate, and trust attorney. They best fit the category of the type of attorney you need.

Your goals are achievable. A consultation with an estate planning lawyer and possibly also a tax lawyer or CPA (depending on the level of tax knowledge of the estate planning lawyer) would be helpful.

There are several ways to do this, you may consider a special type of a trust called an IPUG™ which can separate income and principal during your life and upon your death.

My first question would be why do you want an owner to have equity and no income--is that the best way to accomplish your goal. Assuming it is, you can do what you wish. One of the unique things about an LLC is that earnings do not have to be distributed equally to members. However, remember that this member will be taxed on his share of the income, whether he gets any money or not. In light of that I would urge you to distribute to him at least enough to pay the tax that will be assessed.