My daughter is a huge Blue card fan (she loves mermaids especially). How can I build a solid blue deck on a limited budget with easily obtainable cards?
You should be able to build a reasonable deck of any colour for casual play using just standard commons. It won't be particularly exciting, but as long as you have a reasonably large fraction of creatures, go easy on the enchantments, and choose cards to follow the mana curve (i.e. some low cost, some high cost, most in-between), you'll be fine.
I'm not entirely sure what you mean by 'mermaids'. If you're referring to Merfolk, then yes, you should be able to build a reasonable deck based on that theme relatively cheaply. A basic idea would be 4x Lord of Atlantis, 4x Sunken City, and then your choice of Merfolk cards (of which there are lots; many are common).
Here's a list of Merfolk to give you an idea of the possibilities:
Building an inexpensive blue deck means concentrating on one of the strategies it does well at common, which could mean "mill" (causing your opponent to draw out his or her deck), cheap flyers, card drawing, or creature denial (cards like Unsummon or Sleep). These strategies tend to dovetail nicely with merfolk. Merfolk themselves have a "strategy," as there are many cards that give bonuses to other merfolk you control, or allow you to tap them for various effects. These "fish" decks are playable in many different formats.
A preconstructed blue/white Merfolk deck was released in 2008; they're still fairly easy to find.
I am a big blue fan too, and I have a combo of creatues and instants/sorceries/enchantments. I would lay low on the artifacts, even though most say they are good in blue decks. If you want merfolk, I would try adding x3 Merfolk Sovereign and maybe a Deepchannel Mentor. Also throw in a Talrand, Sky Summoner. You should have some counter-spells like Cancel to hold off till you get a powerhouse like Guile, Frost Titan or Sphinx of Magosi.