(a) providing the best defense of it that you can, and then
(b) by evaluating the strongest objection to it that you can find or formulate, and
(c) reflecting on whether the compatibilist can successfully respond to that objection."
One of the strongest defences of the Compatibilist's argument is a subjective one. (1) All feel as though they have exclusive freedom to choose, unless coerced or restrained. All feel that the choices they make have consequences for which they alone must/may take responsibility and thus regret (for a lifetime) poor choices and rejoice over excellent choices. Yet (2) all feel that they are in some ways constrained and dictated to by environment or by unidentifiable influences that confine the depth and breadth of their freedom to act. This is not to say that these feelings, these perceptions, of this (1) choice and (2) constraint are true or accurate. This is just to say that these subjective feelings are. As a result, the compatibilist philosophy is an experiential one that may confirm its validity through subjective feeling.
A number of theologians over the years have wrestled with this issue and come up with terms such as limited free will and limited determinism to describe or articulate a position which sees these two opposites working together. Religiously speaking, the most important argument for free will is that God created us to be in a relationship with Him and that if he pre-programmed us to love Him, that relationship would be meaningless.
This article sums it up better than I could. However, I will shoehorn my two cents in.
Free will, broadly defined as the freedom to act without restraint (and/or without someone dictating your actions) is a basis for establishing moral codes. If you believe in God, you may or may not believe in Determinism -- all things are planned and you play your part according to an already-written narrative. If you are Atheist or Agnostic, you probably don't believe in Determinism, but you might believe in Fate -- "Nothing I can do about that," you say, and therefore you submit yourself to a higher power, even if that is just nature. We follow rules of society, but we are free to break them if we wish; we are not, however, free to avoid the consequences of our actions. Every action we take has an impact, either on the self or someone else, and so we must think of our Free Will not as carte blanche to act without care, but as a responsibility. We are solely responsible for our own actions, and if we reject that responsibility we descend into anarchy and chaos.
I would guess that one's acceptance/recognition of "free will" depends upon who they are and what ideologies they adhere to. For example, Naturalists did not believe in the concept of free will. Instead, they believed that nature was all-powerful (meaning, more powerful than man). Therefore, nature simply had the last say. Others, who have been oppressed in life may also fail to believe that free will exists. As sad as it is, people who have been oppressed do not feel as if they have the option to do as they wish.
For those who are lucky enough, free will is something which allows us to make decisions, balance choices, and move forward with each. As pohnpei points out, free will allows some to make decisions about some circumstances in life. Given that we cannot control everything, free will remains just out of reach for most (while we like to think that we are in complete control).
The strongest possible argument for this view of free will is that it seems to us to be true. Just to take one example, I have the capability of acting in different ways at this moment. There is nothing that forces me to participate in this discussion and surely there is nothing (other than my free will) that has determined what the next word I type will be.
If we look at a different example, it seems that free will can coexist with determinism. For example, when I fell asleep last night, I did not set an alarm, assuming that there would be nothing that would require me to wake up earlier than I naturally do. That was free will. However, I was wrong. It snowed during the night (determined) and that means I needed to wake up earlier to shovel the driveway. So there is both determinism (the snow) and free will (my choosing not to set an alarm).