The Arkendish can do many things, most of them connected to Thinkamancy.
With it, Charlie can handle an unlimited number of Thinkagrams, and extend some of his abilities to his personal Archons in his tower.
Any commander in the world can get those Archons' attention by concentrating hard enough, for long enough. This could take hours or even days, but Charlescomm would establish contact eventually. This allows him to hire out to any side or barbarian in the world.
The signature unit of the Dish seems to be the Archons, flying female knight-class units with an array of powers. Charlie can pop one archon every turn or three archons every two turns with a turnamancer. Whether this capability is granted by the Dish is unknown. Most other cities cannot pop archons at all and those that can take multiple turns.
Another ability may be to interfere with and/or tap into other communication devices (Eyebook, prime example)Erf-b1-p100 and possibly listen in on other sides thinkagrams [the technology behind the Eyebook]. Immediately after being shown to be "on another call" while Parson gives his Spacerock battle plan to WandaErf-b2-p38, Charlie seems to be acting with knowledge of it when he tries to convince Jillian to return and Tramennis to refuse parley.Erf-b2-p38
The Arkendish apparently enhances (if Charlie is a caster) or grants him the ability to act as a Thinkamancer for the purpose of forming a Caster link over extreme range. In spite of the distance, this link is possibly more powerful than a normal Thinkamancer link.book-2-–-text-updates-014 , Erf-b2-p22 While the caster link is active, the Arkendish can't be used for communication.book-2-–-text-updates-018
Extreme speculation: Mind control/influence, especially when directly communicating. (Maggie is very worried when Parson asks her to contact Charlie.)Erf-b1-p088
More extreme speculation: given its Thinkamancy based abilities, a four-caster link may be possible.
Real World References
The Arkendish resembles the antenna used for satellite television.
It was pointed out that Dish Network formed by Charles W. "Charlie" Ergen provides satellite television service in the United States.