I have mounting holes (0.125 in) at the corner of my PCB project in Eagle, made with the standard "hole" tool. However these are not electrically connected to anything.
There doesn't appear to be a way to make these plated through by specifying a net (unlike a via).
If I want my mounting holes to be connected to the ground plane, do I have to just make an oversize via? Is there some other way?
No, you don't want just a oversized via. Step back and think about the problem a bit. You don't just want a mounting hole, but rather a genuine electrical part. This part should show up in the schematic, be something you place on the board, have a pin connected to a net, and a pad that connection can be routed to, just like other electrical parts.
The correct answer is therefore to use such a part. This would have a schematic symbol with one connection. The package would be a thru-hole pad with the hole size matching your mounting screw and the pad diameter a little larger than the screw head diameter. If you want this to be a ground connection, connect it in the schematic to your ground net.
I have done this before and have a few such mounting hole parts in my library, including one for a #4-40 machine screw.