I vote for a gaming laptop. You may end up in liking that portability and power to play games and do other things on the laptop as well. Otherwise, you would need to have the Host PC on during those hours or days that you're away and when you think you may want to access it from the laptop. What if there's a glitch in the connection, with the gaming laptop it wouldn't matter. I also tend to put my PCs in sleep mode when I'm done for the day, and only do remote sessions on-demand (work bookkeeping) and it's usually within my house from my mid-tier laptop upstairs to my tower PC downstairs.
I also guess how many times are you actually going to be using the laptop for the remote connection, otherwise, a lighter-duty surfing laptop would work.