The earliest dog named Rover I found is in a 1718 list of common names for hunting dogs.
Rover was a common name for a hunting dog in 1718, along with other names such as Bouncer, Fiddler, Gallant, Lively, Ranger, Ruffler, Soundwell, Trouncer, Traveller and Wonder.
The name is most likely from rover, n.2 in the OED, specifically sense 2a:
A person who travels from place to place without fixed route or destination, esp. over a wide area; a wanderer, a roamer; a nomad. Also: an animal which ranges over a wide area.
This is etymologically from the -er suffix applied to rove v.2, the first definition in the OED being:
To shoot at an arbitrarily selected mark, and senses deriving from this.
Etymology: Origin uncertain; perhaps a midland form corresponding to northern rave v.2, either as an analogical formation or as the non-northern reflex of a borrowing < the possible early Scandinavian etymon of rave v.2 (although this would imply a date of borrowing significantly earlier than either English word is attested).
On the other hand, the etymology of piratical rover, n.1 (cf *sea-rover*) is:
< Middle Dutch rōver or its cognate Middle Low German rōver reaver n. Compare Middle Dutch seerōver , Middle Low German sērȫver sea-rover n. and also Anglo-Norman roveres sur le mere , plural (1429 or earlier). Compare later rove v.1
The boy's country-book (1839) by William Howitt features a happy dog called Rover:
The school for children, or A selection of instruction and entertaining tales (1801) "from the French of" Vincent de Langres Lombard features a poem called "VERSES TO MY DOG, ROVER, WHEN GROWN OLD.":
And there's a song in The farmer's boy: a rural poem (1801) by Robert Bloomfield called "THE SHEPHERD AND HIS DOG ROVER":
Which was also printed in The Monthly mirror: reflecting men and manners (1801):
A General Dictionary of the English Language (1780) by Thomas Sheridan may offer an etymology. The definition for ranger gives both "a rover" (presumably human) and "a dog that beats the ground" as synonyms for ranger. Perhaps ranger was first applied to "dogs that beat the ground".
Poetical works (1740) by William Somerville (1675 – 1742) includes "The Officious Messenger, A Tale" which appears to feature a dog called Rover:
Ye world-makers of Gresham-hall,
Dog Rover shall confute you all ;
Shall prove that every reasoning brute
Like Ben of Bangor can dispute;
Rover, as heralds are agreed,
Well-born, and of the fetting breed,
Rang'd high,was stout, of nose acute,
A very learn'd and courteous brute.
With him obsequious Rover trudg'd,
Nor from his heels one moment budg'd;
The trusty Rover lay hard by,
Observing all with curious eye.
The servants, to the stranger kind,
Leave trusty Rover still behind.
Rover, who now began to quake,
As conscious of his foul mistake,
Trusts to his heels to save his life;
It appears in a list of "Hunters' TERMS, &c" in The compleat sportsman (1718) by Giles Jacob, and is given as a common name for a hunting hound: