While doing my research for some good budget-friendly hotels for my trip to Bangkok, I came across some good options. My criteria were family friendliness, proximity to the major attractions and budget-friendly. They are listed here for you:

1. Pullman Bangkok King Power (Near Siam Square)
http://www.pullmanbangkokkingpower.com/

I found this to be the best but could not get reservations here for my chosen dates.

2. Pathumwan Princess (Near MBK Mall)
http://www.pprincess.com

3. Phranakorn-Nornlen Hotel (Oriental Vegetarian Breakfast)
http://www.phranakorn-nornlen.com

4. Century Park Hotel
http://www.centuryparkhotel.com

5. Holiday Inn Silom Bangkok
http://www.ichotelsgroup.com

6. Unico Grande Sukhumvit
http://www.unicograndesukhumvit.com/

7. Siam City Hotel
http://www.siamhotels.com/sch/

8. Reflections
http://www.reflections-thai.com/

9. Dynasty Inn Grande (Near Sukhumvit)
http://www.dynastyinn.com/

10. Amari Hotels (Check out their promotions)
http://www.amari.com

Some more useful hotel URLs you might want to try if you do not find anything that suits you from the hotel suggestions above:

http://silom-serene.com

https://majesticsuites.com/accomodation.html

http://www.admiralsuites.com/

http://www.sleepatseven.com

http://www.shanghai-inn.com/bangkok_hotel.html
In Chinatown

http://www.bossotelinn.com