There aren't a ton of rap lyrics about baseball. This is because baseball is the lamest sport in the world and you and I are the only people who like it. Skimming Rap Genius, I can find more than 100 rap lyrics about LeBron James, but there is literally not a single one that mentions Justin Verlander

So when a rapper does mention a baseball player, it must mean something, and we'd be fools to ignore whatever it means. So here are three of the most important questions in baseball today, as resolved by rappers:
1. Who is the hottest young star in baseball? 
2. Who is the hottest star on the New York Yankees in baseball? 
3. Who is the hottest old white man in baseball? 
NOTE: Because of the nature of this exercise, all videos should be assumed to have profanity and/or material you might consider offensive. This warning should not be overlooked. I really mean it. Sexist, violent, lurid, homophobic, drug stuff, etc. Really offensive!
Query 1: Mike Trout or Bryce Harper or Stephen Strasburg or somebody else? 
Trying to figure out the hottest young star requires a definition for “young,” as three quarters of the league seems young to me because I am old. Buster Posey is certainly youngish, and Cali Elem raps that he’s “tryna be a rookie with a ring like Posey did.” But the bar for young should be lower, so this is going to be limited to players who aren’t yet 25. 
Bryce Harper, for instance, has two rap references, including one by the well regarded Wale, whose Wikipedia profile is 2,700 words long: 

W’s on my mind, peep what I’m getting at
Attire proper, Bryce Harper though slightly darker

and a rapper named LeGit (no Wikipedia profile) also compares himself to Harper: 

I guaranteed knock it out the park, Bryce Harper…

A few lines later, LeGit compares himself to Hitler, so clearly LeGit isn't making a character reference so much as a "knocking it out of the park" reference. Per LeGit, Harper knocks it out of the park.
Wale's line is more nuanced. "W's on my mind" likely has a double-meaning: Both men are single-mindedly focused on winning, and both wear a cap with a W on it. "Attire proper" might mean that Harper, who plays for Wale's hometown team, is playing for the right side, the good guys. "Slightly darker," with the emphasis on slightly, suggests that Wale considers Harper to be, though Caucasian, a relatable person. Wale is vouching for Harper. (The two have broken bread together.) Between Wale and LeGit, we can conclude that Harper has abilities and realness.

Strasburg also gets a Wale reference, along with mentions by Game (5,019-word Wikipedia profile), Skyzoo (903 words), and GR Lynch (no profile). Game’s line name-drops Strasburg without actually naming him: 

San Diego Aztec, number one prospect

while Skyzoo uses Strasburg for a pretty simple pun that I don’t quite get: 

Stephen Strasburg your curve/ Let a base load

And Wale boasts that he will “throw around game, Strasburg pitching.”  I assume it’s “throw around game,” at least, and not “throw a round game” as it appears on various lyrics sites. "Throw a round game" isn't a phrase that seems to exist. Wale seems to say he has game and distributes it widely with great ability. 
There are no Mike Trout references in the genre. If any rappers are members of the BBWAA, they likely voted for Miguel Cabrera for MVP, but there are probably no rappers who are members of the BBWAA. There are also no references to Giancarlo (or Mike) Stanton, to Jurickson Profar or Manny Machado or Oscar Taveras or even Justin Upton
Rap battle winner: While Strasburg has the most references, his name seems to be invoked more out of convenience. Harper’s references feel more personalized. Bryce Harper wins the rap battle!
Both players show up often. Rodriguez is mentioned in about 20 songs, his surname rhymed with “Adidas” and “road to riches” and his nickname rhymed with “day job,” “based God,” “AWOL,” “game all,” “stay hot,” “applaud” and “they not.” Lil Wayne matches “ARodding” and “headlining” in an internal rhyme, because of course he does.  
Jeter is named in around 30 songs, rhymed with 
  • Millimeter 
  • bleedin' 
  • tweeter 
  • skeeter-skeeter
  • cheaters 
  • see a
  • fecal 
  • G'd up 
  • sneaker 
  • heater 
  • feeder 
  • meet her 
  • Aretha
  • beat her 
  • demeanor 
  • ringleader 
  • parking meter 
  • FIFA
But while each player is mentioned a lot, each player is also mentioned in a particular way. Rodriguez is noted for his money, his dating history, his scandals. Rappers treat him as a celebrity, basically, a famous tabloid figure with public excesses. “My material girls give 'em A-Rod moneyyy,” says Wale in one song, and “wanna see Ross get the A-Rod loot” in another. Lil B (574-word Wikipedia profile) mentions A-Rod’s money, though he undersells it by referring to Rodriguez’s “10 mil contract.” Chris Webby (no profile) complains that A-Rod is overpaid:  

But I put out more hits than A-Rod
And gotta work a seven-dollar an hour day job

“You my stand-in Cameron, let me be your A-Rod,” says Childish Gambino (2,118 words). Common (3,462 words) brings the two themes, the money and the starlets, together: 

Whatever you want, you could quit your day job
You could be my Kate, I could be your A-Rod

And his steroid use comes up, too. Money Making Jam Boys (no profile, but a side project of the Roots’ Black Thought) invokes A-Rod to rhyme “Balco buddy” with “Kid Cudi.” And Slaughterhouse (768 words) zings him. Zings him real good.

This fiscal year I'mma stay hot buzzin
With dudes that help me shoot like A-Rod cousin

Jeter seems to get more respect. As a rap protagonist he is defined by his role on the Yankees more than his role in the tabloids. From Ghostface Killah, for instance, 

Ringleader set it off, rap Derek Jeter

a line that paints Jeter as a winner. Diggy Simmons specifically compares his role in rap to Jeter’s role on the Yankees: 

I'm the lead-off like Derek Jeter

while Cam’ron (among other rappers) notes that Jeter plays the crucial position on the diamond: 

I'm Derek Jeter, cause I'm in between the base my mane
And thats on Lennox, 7 for 8th ace of kings

While ballplayer accomplishments are often used for non-specific pun-based boasts (“I hit hos like Derek Jeter”—Dougie D), Jeter’s accomplishments are actually named specifically. His 3,000th hit, in particular, got rap recognition. Jadakiss:

I could do it 1,000 times.
Mattera fact I could do it 3,000 times.
Call me Mr 3000
Jada Jeter.

Shoot, Skillz (665 words) likes Jeter so much he gives him credit for the all-time home run record: 

Three thousand home runs, Jeter's the dude

There are two extra bits that make this an easy victory for Jeter. One is that he is a New York icon to New York icons: Method Man (3,450 words) and Ghostface Killah (1,712) both rap about him, as do notable New Yorkers Cam’ron (2,430) and Foxy Brown (2,233), Russell Simmons’ son Diggy Simmons (1,014), and former Notorious B.I.G. protege Jadakiss (1,681). Among New York heavyweights, Rodriguez can claim only Jay-Z (6,051 words), whose bar about Rodriguez actually came when Rodriguez was still a Texas Ranger. Rodriguez might have the wealth and status to get rappers’ attention, but Jeter is the one New Yorkers actually love. 
The other is the J. Cole song that mentions both in the same line:
and clearly positions Jeter ahead of A-Rod: “Is that your girl? Well, I just G'd her, no A-Rod.” From Rap Genius: 

To “G” someone’s girl is to take the girl away from them, gangsta style (G stands for gangsta)

Derek Jeter (sounds like “g’d her”) is a teammate of Alex “A-Rod” Rodriguez on the New York Yankees

He is also referring to when A-Rod was traded to the Yankees he did not take Jeter’s position as the Short Stop. (Since both Jeter and A-Rod originally play the same position). He’s inferring that no matter how good you are, you aren’t taking his spot.

Derek Jeter wins the rap battle!
Query 3: Bobby Cox or Tommy Lasorda 
If you are discouraged because you have never been mentioned in a rap song, take heart: neither has Seattle Mariners manager Eric Wedge! In fact, unsurprisingly, the default for managers is no mentions: Bruce Bochy, Sparky Anderson, Whitey Herzog, Mike Scioscia, Jim Leyland, Davey Johnson, Earl Weaver, Terry Francona, Clint Hurdle, etc. are all shut out of hip hop. A New York connection helps, and Joe Torre, Joe Girardi, and Don Mattingly each get a mention or two, but everything in New York gets brought up in a rap lyric at least once. There are three managers who got rappers’ attention without being in New York. 
Bobby Cox is known for getting ejected. The Rapper H (no profile):  

Bobby Cox I throw that base and now the majors popping off
Sophisticated wordplay, b****, add a homophone

Among the countless base/base or base/bass punchlines in rap lyrics, this is probably the best. It’s a much more convincing nod to Cox than Cyhi Da Prince’s (348 words) more generic line: 

I'm shy as hell, no Lupe
I'm Bobby Cox, too Brave

Tony La Russa gets the highest-profile mentions, by Jay-Z and Fabolous (1,605 words). Both lyrics are more about the St. Louis nine’s club name than about La Russa’s particular attributes: 

They say its celestial, its all in the stars
Like Tony LaRussa on how you play your Cards


But I ain't into coaching birds like Tony LaRussa

Lasorda, meanwhile, is named in no fewer than seven songs. Sure, most of the lyrics take advantage of his name—what else rhymes with “Michael Rappaport,” Dom PaChino (393 words) must have wondered for most of an afternoon—or the action-verb qualities of his team name: 

Huh, yo Tommy Lasorda
Dodge bullets, dive into the water

But seven rap mentions is seven rap mentions, and establishes Lasorda as the old white manager with, by far, the most rap credibility. Tommy Lasorda wins the rap battle! Wale, rap’s biggest baseball fan, gets the final say: 

I'm sort of a genius, nothing short of a legend
Sort of Tommy Lasorda
The way I'm sorting these pitchers
Pictures whatever.

So now that’s done. 
Lyrics/explanation of lyrics mostly possible thanks to