The Spiders are more than just their 1899 season, but that's the single most famous thing they are known for. I wouldn't want my team to be called the Spiders.
If I were running CLE, I would be pushing for Fire or Flames. Comes with a built-in brand and aesthetic. You can wear fire department colors, you can have a dalmation as a mascot, you can play a fire siren when the team scores. Hell, the bullpen cart can be a fire truck.
If they want to go the Negro League route (as, I understand, the Nationals almost did), the most established Negro Leagues team in Cleveland were the Buckeyes.