Julian Lennon had a nice album as i recollect.

ben taylor

Wilson Phillips (I am not interested in them, just mentioning them)

*Carnie and Wendy Wilson and Chynna Phillips

(We will not speak of Miley ...   that goes without saying)

Jakob Dylan
Derek Trucks (uncle)
Nancy Sinatra
Jeff Buckley
Arlo Guthrie

Dino, Desi and Billy were sons of Dean Martin and Desi Arnez (Lucille Ball's husband).  Billy was a family friend.  This group had a couple of charted hits.  The fathers were far more notable.  Dean Martin was a huge all around great entertainment star.  Desi Arnez conducted a Latin jazz/pop orchestra and sang.  He also was a big TV star.

Jesseyeric, this is a great topic.  There are many more 2nd generation musicians, in all of music, but I will give others a chance before I respond again. 

I have all of Julian Lennon's work and I think he is very talented.

He suffered from lack of good promoting, in my opinion or too many people tried to hold his work up against his dad's songs ( talk about a tough act to follow!). As he is not Yoko's son, I understand he got absolutley no advice or help form her, which may have been a good thing!

Arlo Guthrie is a great folk singer and he made some cool songs, also he is very interesting to listen to in interviews.

I don't know if Gary Lewis counts, as his dad , Jerry was a comic actor, not a musician, but his group created some excitement in the late 60's and he had a few hits that were kind of nice little pop songs.

• father: Mel Tillis - daughter: Pam Tillis (both charted country)
• father: Nat King Cole - daughter: Natalie Cole
• father: Ricky Nelson - sons: Nelson Brothers (charted early 90's)
• father: Hank Williams - son: Hank Williams Jr.
• father: John Mellencamp (charted "Let It All Hang Out") - son: John Cougar Mellencamp
• Classical Music: Johann Strauss, Richard Strauss ... there could be more.  I'm not sure how they were interrelated.
• mother: June Carter Cash - daughter: Carline Carter ... I'm a little fuzzy on the intrrelationship. (both charted country)
• parents: Ozzie and Harriet Nelson (40's big band) - son: Ricky Nelson
• mother: Judy Garland - daughter: Liza Minnelli
• father: George Morgan (charted country) - daughter: Lorrie Morgan

Note: all the above charted on a national level 

Cavsfan

Nelson? You just killed my holiday

In Response to Re: 2nd Generation Musicians:

Maybe I can make up for it by finding more national musicians that are related. It will be harder. Maybe there are a lot more out there. At some point we have to be running out of relatives.      

• father: Eddie Levert (O'Jays) - sons: Gerald and Sean Levert (R&B group: Levert)
• father: Bob Marley - son: Ziggy Marley
• "Mother Maybelle Carter" (part of Carter Family band that's in the Country Music Hall Of Fame) - daughter: June Carter Cash 

In Response to Re: 2nd Generation Musicians:

Very good.  Arlo Guthrie's dad was Woody Guthrie, a great songwriter and folk singer.  The great folk song we probably would remember is: "This Land Is Your Land" ( '........ This land is my land.  From California, to the New York Islands .......... This land was made for you and me.')  

Great topic Jessey.
Ringo Starr's son Zak Starkey plays drums for Roger Daltrey/Pete Townshend (The Who) and I believe he's still with Oasis.He might have played for Ringo Starr and his All Starr band at some point as well. 

Father: Billy Ray Cyrus - daughter: Miley Ray Cyrus.  Miley is a pop singer and actress.  She plays Miley Stewart/ "Hannah Montana" on The Disney Channel.  Miley has recorded songs through Disney and other labels.  Her dad Billy Ray had a couple of big country hits.  "Achey Brakey Heart" and "Some Gave All".