Can I increase the number of allowed saves?

Well, hopefully you'll find a better answer, but one workaround is to simply back up some of your older saves to another directory.

On Windows 7, my saves are stored in

C:\Users\MyUserName\Documents\BioWare\Mass Effect 2\Save\John_12_Soldier_260110

That last bit differs on a character-by-character basis. You could grab the .pcsav files from there and move them to a backup directory. Now you've got some archived saves and free slots. If you want to restore them later, you just drop them back in (though you probably need to make sure you aren't bringing the total over 50 if you do).

Epic Legion has created a program that will remove the limit of 50 saves on the PC version of Mass Effect 2. It also can open the ingame console plus other tid-bits of cool stuff if one so desires.

The program is called the PCC PATCHER
located here.

PLEASE NOTE: You must have the version 1.02 ME2 Executable for this to function.

Here is how to open and use its functionality:

1. Download & extract the enclosed folder from the linked zip file to your Mass Effect 2 Binaries folder: Usually located here: Program Files\Mass Effect 2\Binaries
2. Open the PCC PATCH folder and drag the enclosed files directly into the ME2 binaries folder.
3. Double click the "PCCPatcher.exe" that is in the binaries folder now and then proceed to the below instructions once the program has opened:

In the open PCC PATCHER PROGRAM

1. File > Open engine
   usually located here: Program Files\Mass Effect 2\BioGame\CookedPC
2. File > Open SFXGame
   usually located here: Program Files\Mass Effect 2\BioGame\CookedPC
3. Basic Mods Tab
4. Select the tick boxes that you want for your game.
5. Main Tab > Save Packages
6. Run the newly unofficially "Patched" Mass Effect 2 and enjoy the enhancements.

This is not mine - the credit goes to phoenixofthunder. I just copied the above from the official ME2 forums.