I found another website with lots of abstract strategy games that take place on grids, so many of them are playable with MOD64. Here is a link to the website. The website also allows you to play these games online. If anyone finds any nice games that work well with MOD64, it would be awesome if you could post it on this forum!
Games that are potentially playable with MOD64:
9tka
Akron
Amazons - Try different board sizes and shapes, and different numbers of players
Reversi and Anti-Reversi
Arimaa
Ataxx
Bashni
Breakthrough
Cam
Camelot
Cannon
Chain Lightning
Clobber
Connect 4
Connect 6
Crossway
Dama
Dameo
Dipole
Emergo
English Checkers
Epaminodas
Fields of Action
Flume
Forms
Fox and Hounds
Freedom
Go
Gonnect
Grand Chess
Halma
Hanniball
Harzdame
Hawalis
Hnefatafl
International Checkers
Jeson Mor
Jungle
Cowrie
Konane
Lasca
Lines of Action
Ludo
Mad Bishops
Men Row Chess
Metamorphosis
Mind Ninja
Momentum
Monkey Queen
Murus Gallicus
Neutreeko
Ordo
Oust
Oware
Period 5
Phutball
Pilare
Quartetto
Quax
Regoot
Rekushu
Renju
Rhode
Rush
Russian Checkers
Shakti
Shatranj
Shintai
Shogi
Slimetrail
Slither
SnailTrail
Speed
Splut
Stavropol Bashni
Stavropol Checkers
Stilts
Suicide Bashni
Suicide English Checkers
Suicide Russian Checkers
Surakarta
Talpa
Tama
Tanbo
Tavreli
Teeko
Three Musketeers
Toguz Kumalak
Traffic Lights
Twixt
Ultimate tic-tac-toe
Vai Lung Thlan
Veletas
Virus Wars
XiangQi
Yugo
Zurero
Whew, that's a lot of games! Some of them require a lot of pieces or might require some innovation to make work, but a lot of them should be playable with the base set of MOD64 and not too much work.