I would live in the Midtown or Houston Heights areas of Houston. They are both pretty central and have tons of great bars, restaurants, and parks within walking distance. Lots of single professionals living in these areas and a great night life. Both areas would meet your requirement for the 25 min drive to Willowbrook Mall (assuming non-rush hour traffic times).

I 100% agree with all the neighborhoods the others have listed. I lived in Houston for 6 years during my 20s and those are your best bets -- close to the action, lots of character, and relatively safe. However, you mention you need to live near Willowbrook -the problem is Willowbrook is a 30-minute drive from all those cool neighborhoods inside the loop. Willowbrook does have some nice neighborhoods nearby - esp Vintage Park, Champions, and Jersey Village - but they are much more family oriented and the dining/restaurants are not great. If you do decide to live on that side of town, I can't recommend Raveneaux Luxury Apartment Residences highly enough.

There are several apartment complexes, town homes, and neighborhoods around The Vintage - Willowbrook | Kelsey-Seybold Clinic   It's minutes from Willowbrook Mall. 

Look inside the 610 loop near Washington Ave./ Memorial Park or just outside the loop in the area surrounding the Galleria Mall.

I lived in Houston for most of my life. I recommend the Houston Heights, Rice Military, Montrose, Midtown, and the Greenway/Upper Kirby Neighborhoods. Downtown has been improving as well. The best bars and restaurants in the city are located in these neighborhoods. The two main city parks, Washington Ave./ Memorial Park and Buffalo Bayou Park, are in the vicinity as well. 

My top 3 recommendations are the Heights, Rice Military and Midtown. Heights is more of a neighborhood with small houses and townhomes. If you are going to live in Midtown or Montrose it will probably be in an apartment/condo. 

Good Luck!

Firstly, if you want to live in or near an area frequented by young people, you're going to have to be "inside the loop," meaning with in the 610 loop around Houston. This is probably going to prevent you from being 25 minutes from the Willowbrook Mall during any hours with traffic (7am-9am, 3pm-6:30pm on a good day).  

If you're willing to fight that, though, here are your best places to live in Houston for your criteria: 

Plenty of Restaurants and Bars:

1. Washington Corridor along Washington Ave. The center of this would be around the intersection of Heights and Washington. There's no heavy concentration of bars or clubs in Houston where you'd be able to "bar hop" like you would on 6th Street in Austin, but this is the closest you can get to that experience. 
2. Also good is West University Place, where there are lots of students but also young professionals and some good bars and music venues along Morningside Dr. There's also a nice running trail around Rice. 
3. East Downtown, or "EaDo," near the BBVA Compass Stadium has a budding nightlife scene, some good places to eat, but no good grocery stores or anything practical. 

Running Trails

1. Bayou Park just went through a huge expansion and renovation, and is a fantastic park.  If you run the full loop from downtown to Shepherd it is about 5 miles. Also close to downtown and the Washington Corridor nightlife. 
   
2. Washington Ave./ Memorial Park is always a favorite. There is a 3 mile loop and a fair amount of wooded trails good for mountain bikers. Again, this is close to the Washington area. 
   

General Areas of Young Populations

1. Houston Heights has exploded in the past 5 years, as have costs and property values. Very cool feel and lots of "local" joints. Different areas have different feels, but all are attracting the young professional demographic. 
   
2. Greenway/Upper Kirby is a good area, too, and close to both Rice and Allen parkway/Bayou Park. 

You should have a look at the Washington Corridor area of Houston.

Houston Heights is a great place to live but may be further than you want. The Woodlands would be a good choice.

I would say the closer you are to Willowbrook the better. There are a few bar and restaurants within the mall area and some further on down 1960. Houston is very spread out so a 25 minute drive may not get you as far as you'd like.

Midtown meets all your requirements!