After Beren slew the party of Orcs that had killed his father, Barahir, and their band of outlaws, and reclaimed the Ring of Finrod (as it was called before being renamed), Morgoth placed a price upon Beren to the equal of that of a High King of the Noldor. Was Beren that powerful of a man to attract the attention of Morgoth?
Beren was prophesied
Beren should be considered among the greatest of the Men to have graced Middle-earth. Especially considering his feats later in life.
While Beren is being hunted by Sauron's army after having a bounty placed on his head, Thingol declares that no Man shall enter Doriath, not even that of the House of Bëor
'Into Doriath shall no Man come while my realm lasts, not even those of the house of Bëor who serve Finrod the beloved.' Melian said nothing to him at that time, but afterwards she said to Galadriel: 'Now the world runs on swiftly to great tidings. And one of Men, even of Beor's house, shall indeed come, and the Girdle of Melian shall not restrain him, for doom geater than my power shall send him; and the songs that shall spring from that coming shall endure when all Middlle-earth is changed.'
The Silmarillion: Quenta Silmarillion, Chapter 17: Of the Coming of Men into the West
Beren's fate was directly connected to that of the Silmarils. It was the bounty and fleeing from Sauron's army that lead him to Doriath to meet Luthien. We must recall that the Girdle of Melian was impenetrable, and only through great need or desperation (such as Beren's) can one pass through it, only when the doom is greater than the power of Melian the Maia to keep them out. Melian talks of songs to be sung about Beren long after Middle-earth is changed (possibly foreseeing the destruction of Beleriand and the raising of Númenor)
Beren was renowned during his time as an outlaw and greatly feared.
It is clear that Beren's renown had grown great in his time in Dorthonion avenging his father, and even without the prophecy of his coming and his greatness. Orc bands feared him and Morgoth was forced to send an army, commanded by his top lieutenant, after a lone outlaw.
[...] but the Orcs fled rather at the rumour of his approach than sought him out. Therefore an army was sent against him under the command of Sauron; and Sauron brought werewolves, fell beasts inhabited by dreadful spirits that he had imprisoned in their bodies.
The Silmarillion: Quenta Silmarillion, Chapter 19: Of Beren and Lúthien
The need of Morgoth is evidently great if he is to send an army after a lone outlaw. The threats posed by Beren and the fear instilled in the orcs gave Morgoth great need to dispose of him, setting the same price (or possibly higher) than he'd set for the High King of the Noldor.
In the earlier uncompleted Lay of Leithian, the writing of which was mostly abandoned in 1931, we get another hint of the bounty of Morgoth and the threat Beren, and on this occasion his companions, posed on Morgoth and his orcs.
Such deeds of daring there they wrought
that soon the hunters that them sought
at rumour of their coming fled.
Though price was set upon each head
to match the weregild of a king,
no soldier could to Morgoth bring
news even of their hidden lair’
History of Middle-earth 3: The Lays of Beleriand - Lay of Leithian, (around line 175)
This time it would seem because of the prowess of the band of twelve in Dorthonion Morgoth was forced to put a bounty on each head of the band, yet no one was able to discover their whereabouts until the betrayal of Gorlim the Unhappy, one of the Outlaws.
It would seem this version was not used for the final Silmarillion, with Toklien likely having changed his mind (as he was like to do) between 1931 when he mostly gave up the writing of the Lay and his dead in 1973.
Morgoth must have heard enough about Beren to warrant such a high bounty, given they had never met, but there was at least enough myths and legends about Beren at the time.
Morgoth wanted to be sure
As far as canon is concerned, we have only this:
Thereafter for four years more Beren wandered still upon Dorthonion, a solitary outlaw; but he became the friend of birds and beasts, and they aided him, and did not betray him, and from that time forth he ate no flesh nor slew any living thing that was not in the service of Morgoth. He did not fear death, but only captivity, and being bold and desperate he escaped both death and bonds; and the deeds of lonely daring that he achieved were noised abroad throughout Beleriand, and the tail of them came even into Doriath. At length Morgoth set a price upon his head no less than the price upon the head of Fingon, High King of the Noldor; but the Orcs fled rather at the rumour of his approach than sought him out. Therefore and army was sent against him under the command of Sauron; and Sauron brought werewolves, fell beasts inhabited by dreadful spirits that he had imprisoned in their bodies.
- J. R. R. Tolkien, The Silmarillion, Of Beren and Lúthien
Beren had been in hiding for a very long time, 4 years. During which time he had evaded the servants of Morgoth and they would flea rather than approach Beren at this point, so there's definitely something worthy of note about Beren. Beren was also 'friendly' with the natural things of the land, perhaps he appeared greater than he was.
Morgoth had a sense of Beren's importance
Of the narrative and hidden power of Beren, Tolkien writes:
...the first example of the motive (to become dominant in Hobbits) that the great policies of world history, 'the wheels of the world', are often turned not by the Lords and Governors, even gods, but by the seemingly unknown and weak – owing to the secret life in creation, and the part unknowable to all wisdom but One, that resides in the intrusions of the Children of God into the Drama. It is Beren the outlawed mortal who succeeds (with the help of Lúthien, a mere maiden even if an elf of royalty) where all the armies and warriors have failed: he penetrates the stronghold of the Enemy and wrests one of the Silmarilli from the Iron Crown. Thus he wins the hand of Lúthien and the first marriage of mortal and immortal is achieved.
-J. R. R. Tolkien, Letter 131 To Milton Waldman
Melkor was, before being corrupted into Morgoth, the chief among the Valar, he was the most connected to Eru and was supremely 'powerful' and insightful. He may have felt the power and danger in Beren even before the events that turned Beren into the 'One-hand'. Thus he may have set this high price to be sure.