So Id like my next fire arm selection to be 556. I decided I would go with the Mini 14, with troy stock. People say they have alot of problems, others swear by them. One thing that is irrefutable is the barrel. I'm looking for a good urban brawler here. The stock barrel is not enough to last even one fire fight.

Accuracy systems makes excellent barrels for the mini 14. Lengths are from 16 to 24, I think the 16 would be the obvious choice since I am not looking to reach out and touch.
Here is where my question is. The barrel thickness is offered in these diameters,   1"  .875"  .750"  and .625". Which of these would be best for a tactical weapon? Should I just go with the inch or would it be too heavy?

The one inch would be heavy as hell.  Mini's are already a bit nose heavy because of the long stroke piston system.  Think AK type balance.  If I were going to make a varmint rifle, the 1 inch would be a good choice, but not for a fighting carbine.  I'd probably go with the .625".  Hell, I'd strongly think about SBRing it down to 14.5".  Or if you don't want to the Form 1 song and dance, get a long flash hider permanently attached.

A lot of people use lightweight barrels on fighting ARs, though, and they seem to do OK.  Although I think the profile of lightweight AR-15 barrels in 0.625" as opposed to the the Mini, which I *think* i something like .562".  It's really thin which is chiefly why it gets hotter than hell so easily.

I would like a shorter barrel, but he only goes as short as 16, and I would want to do another form 4 anyway.

ASI does outstanding work.  I'd find a Mini with a 16" barrel and handle to see what you think of the balance.  Then ask Carl at ASI how much heavier the .625 barrel is.  Proceed from there.