What are my legal rights if we had a contractor quit on a home repair job which is now unfinished?

If you already paid him any amounts in excess of the work he's done (e.g. you paid for 100% work up front, but he only did 50%), you can sue him to recover the balance of the excess paid (in this example, if you paid him 100% but he did 50% of the work, he could keep 50% while returning the other 50%).

If you have to hire someone else to complete the work and because the first contract left things half done, it costs extra for the new contractor to complete it--more than it would have--you can sue for the excess, too. So, for example: say that if one contractor should be able to do the job start to finish, for $20k.  Say you paid contractor A $10k and he did 50% of the job before quitting; say for contractor B to pick up the job and fix it will cost anothehr $15k, because he has to rip out and redo some work which is not done the way he would have, so the total cost is now $25k, you could recover the extra $5k.

If the half-finished work caused damage, you can recover for that damage: say that the contractor opened the roof but did not seal it up, so water got in and ruined your kitchen cabinets to the tune of $8; you could get that $8k of compensation.

What you can't effectively do is get a court order requiring the contractor to finish the work; in cases where others could complete it (performance is not unique to that person, the way, say, Scarlett Johanson's performance as "Black Widows" in an Avengers movie is unique to her, and the studio couldn't really replace her mid-filing), the court doesn't like to order someone to do work; they'd rather pay you, as per the above, for any losses or extra costs you incur.

Note, though, that if the reason the work stopped is the contractor went out of business--especially if the contractor is not a person, but was an LLC or corporation which is now either defunct or still in existence but not operational--then even if you sue and win, you might not get any money; if the party sue has no money, you can't recover as a practical matter no matter what the court says.