I will be taking my 8-year-old daughter to NYC for the first time. I am excited about taking her to a city that I absolutely love, but am limited to about 24 hours. We will be arriving via megabus on Saturday at 6:45 p.m. and leaving on Sunday at about the same time.

Suggestions for places to eat, please!!! No chain restaurants.

I have pre-purchased tickets for the Statue of Liberty and the Empire State Building (her choices). Any other suggestions? We will also be spending 4 days in Washington DC, so... we will be going to the National Zoo and Museum of Natural History there prior to seeing the Big Apple.

I was considering trying to catch a musical... (which would be her first)... she absolutely loves music and dancing. Is there a chance of catching a deal at the TKTS booth with only an hour before the musical starts? We will be arriving at Penn Station at 6:45. I was hoping to see MARY POPPINS.

I am prepared for a reality check. Please help!!! : )

Where are you staying? The answer will help determine whether you'll have time to get to TKTS before theater time. (I assume you will have a bit of luggage since you're going on to Washington. So, going to the theater before stopping at your hotel may not be an option, since, I believe you will not be able to check luggage at the theater.)
One thing in your favor is that there usually is no line at TKTS after 7 p.m. On the negative side, however, Saturday is the busiest night for the theater. And, there aren't that many musicals appropriate for an 8-year-older. In fact, the only two that MIGHT (and I emphasize the word "might") be at TKTS are Mary Poppins and The Little Mermaid. On balance, I would not get my hopes up.
What's a better idea is to go to broadwaybox.com and check out the discount tickets for those two shows and get the tickets in advance. They won't be as cheap as TKTS, but the are both available on discount.

We will be staying in the Theatre District... on W. 45th St. I think the TKTS booth is only 2 blocks away.

What time does Mary Poppins start? 8pm? I would be leery to buy tickets in advance for a show on your arrival day, esp if your bus isn't scheduled to arrive until 6:45pm. If you are lucky, you will arrive at 6:45pm, but if you're unlucky (traffic can be bad heading into Manhattan on Sat evenings) your bus may end up being 15 or 30 minutes late. So now you're talking about arriving after 7pm, go to your hotel, check-in, and get to the theatre by 8pm? Unless your hotel is right next to the theatre, I really would not recommend this in case your bus arrives late and you end up missing (part of) the show.

Just an example, I took Megabus from Boston to NYC a few weeks ago on a weekday afternoon. The bus was late to depart Boston, and ended up arriving late for over 30 minutes.

For a fun place to eat go to Ellen's Stardust Diner. Basic diner fare but the waitstaff is Broadway chorus boys and girls between shows and their performances are a lot of fun. This might solve the problem of coming in awfully late to be sure of getting to a show Sat night.

I assure you -- even if you don't get to see a show that night, she's going to be so wide-eyed at the sight of Times Square (sorry, nyt!) that it's not going to matter. Have dinner and wander around the 'hood!

If you don't make it in time for a Sat. night performance of a show, Mary Poppins has a Sun. matinee at 1 pm that would be out by 3:45. There are not discounts to pre purchase for this Sunday, but you might find discounts at the TKTS booth in Times Square which opens at 11 am on Sun. morning. People start lining up much earlier though. I've gotten there at 10 and found it mobbed. Your other option is to buy full priced tickets now.

I have to say you've picked the 2 most time consuming sights to visit (ESB and SOL/Ellis Island) for a 1 day trip. Top of the Rock, the observation deck at Rockefeller Center is another great spot. Trattoria Trecolori is a popular Italian restaurant in the Times Square area. If you don't go to a show on Sat. night, you might head to another neighborhood like Greenwich Village and eat there. What kind of food do you/she like?

I would take my eight year old to central park, get a carriage ride, have a look at the Plaza(Home Alone), check the street art, maybe have lunch at the Hard Rock or the Boat House.... have fun!