Backyard Aquaponics
A place to discuss aquaponics

Would it work if I have a 12 volt 20 ah battery, a 40-60 watt solar panel, a 300 watt inverter for an ac 17.5 watt water pump?

That depends on how long you plan to run the pump for each day.

For 24 hour operation, assuming (optimistically) 80% inverter efficiency, that will be a bit over 500Wh per day. A 60W panel with good sun exposure in Phoenix should average about 350Wh per day with an MPPT charge controller, so you can see that on average it isn't enough. In summer it will be enough most days if there isn't too much cloud around, however, your battery will be run rather flat each night, so it wont last very long. For most of the year it would be hopelessly inadequate.

Is there a cheap solar panel that is 80 watts for cheap? And controllers and stuff what wires are needed for this Connecting the solar panels to the battery and to the controller and the inverter?

Ebay should answer that question nicely: http://www.ebay.com/sch/i.html?_from=R40&_trksid=p2050601.m570.l1313.TR0.TRC0.H0.Xphootovoltaics.TRS0&_nkw=phootovoltaics&_sacat=0
I'd shoot for much more wattage coming in, and bigger and better batteries too. @Gunagulla knows what he is talking about, he could give you the numbers you need.

A smaller inverter (modified sine wave) running closer to it's capacity would be slightly more efficient.

My limited experience is that we ended up with 3X as many panels as we "calculated" due to clouds, wire losses, and the drain from the inverter even when nothing was running off of it.

And is wiring the whole system difficult? It seems very daunting. And as a note I do have 220 dollars for the whole solar panel system.
What about a cheap and good quality controller? Wiring for the solar panel complex? Small inverter? Battery size (35 ah)?

60W is hopelessly inadequate, 80W isn't much of an improvement!

$220 might struggle to cover a suitably sized reasonable quality system with new parts, but you might do it with 2nd hand gear, but I would recommend not buying a 2nd hand battery. You'll want a battery of about 100AH, a 15A charge controller, suitable PV panels -voltage and current to suit the controller, at least 120W.
An MPPT charge controoler will give you a bit more output from your panels than a PWM type will, so with PWM you would need a bit more PV power.

Cheap inverters are notoriously inefficient, and some use quite a lot of power even with no load, so you need to take that into account.
Also, you probably should have a way of charging the battery in case you have an extended period of cloudy weather.

Is there any cables I will need?

Yes- PV panels to charge controller, charge controller to battery, battery to inverter, inverter to pump, all the DC cables need to be sized suitably for the load, and you probably should put a fuse between battery and inverter, if it doesn't have its own input fuse. A fuse would be useful between charge controller and battery too.

If you want it to be electrically safe, you should use an earth leakage device (safety switch, you can buy power boards with them) on the AC side, so you cant get electrocuted around the FT, but you should get someoone who knows what they are doing to set it up properly.

If you use a pure sine wave inverter the pump should run smoothly, but the modified sine wave inverters can make some motors run very roughly, and may damage them in the long term.

do a search on this site for LEAP and low energy aquaponics.

There are some ways of making a system worch with a lot less energy by using air lifts which might be worth a look before going solar.

Any examples of LEAP/low energy aquaponics? As this is an IBC tote system.

Any good 12volt DC water pumps? Wires done already?

search.php?keywords=12v&terms=all&author=&sc=1&sf=titleonly&sk=t&sd=d&sr=topics&st=0&ch=300&t=0&submit=Search

Let the search engine be your friend.

Would a timer consume a lot of power? And is it possible to run the pump for half the day?