I'm thinking preppy wicker here... Paint the bases glossy white, and cover the tops with some padding and brightly patterned fabric. http://www.etsy.com/listing/47625449/amy-butler-full-moon-polka-dot-lime-1

I am in love with blue/orange combo right now. I would spray paint the bottom navy blue and use an orange fabric to cover the top, like http://www.bandjfabrics.com/servlet/the-1819/cotton-lawn-print-white/Detail.
You can purchase rollers/wheels for the bottom to add some height and then they can move around easily!
Maybe add some foam under the fabric for comfort...?

oops, link to the fabric might not work.
B and J fabrics, search number 1139, it's a cotton broadcloth print in orange. have fun!

It looks like there aren't any bottoms to those, so how about flipping them over so the laminate is the base, covering it with fabric, and putting a wood top or even a glass top (depending on the age of your kids) on top of the bins to create a little table?

I was gonna say paint the base a color and give them a thick glass top so that you can see the interesting base but if you think the glass will be a hazard for your son, maybe not. :) I like the idea of turning them into mini-ottomans with a padded top and casters to roll. Having kids myself I would suggest going with an easy to wipe down smooth faux leather/vinyl top and painted base to coordinate with your decor.

I don't know why but I see the bottom painted fushia and the tops heavily padded and upholstered in a graphic orange and white pattern. You should have lots of fun with this one ;)

I agree that padded fabric tops would look great, but bear in mind that you can also paint laminate-- just make sure you sand a bit first and then use a good quality primer so that your paint sticks. This could look awesome with the top painted in an interesting way and then topped with glass (plexiglass so it doesn't break when faced with your son?). I love this find!

A piece of clear lucite (as thick as the old top) could be very nice on top. Leave bottom as is.

I like the idea of turning them into ottomans by putting foam on the tops and then covering it in fabric. I agree with others that spray painting the base a bright, glossy colour is the way to go. Here's a link on turning a coffee table into an ottoman: http://curbly.com/modhomeecteacher/posts/3749-how-to-turn-an-ugly-coffee-table-into-an-upholstered-bench

The other option you have is to paint the table tops with melamine paint. Melamine is pretty toxic stuff so be sure to ventilate well. Here's some instructions: http://www.hgtv.ca/articles/articledetails.aspx?ContentId=1915&cat=1&by=3

You could paint the laminate (as mentioned, with sanding and a good primer, first) maybe even the same color that you choose for the bamboo. Then you could decoupage wallpaper designs or paper cutouts or some other fanciful designs on top of the laminate, then place a glass or plexi tabletop over both units, either close together or spread apart. (Children may be prone to sitting on anything this height, though, so anticipate that!)

I'm envisioning a bright accent color with bright decoupage of parrots or tropical flowers or something fun and colorful. But it would also work well with a black lacquer look and muted Japanese-looking designs with metallic gold in them, for elegance. Or maybe navy blue with a different sort of motif...

I'd reconsider the legs. I think you would need to build a platform base to attach the legs to, and then fasten the cubes on top of the base for that to work structurally, which might or might not look good. (If you do this, paint the base before attaching, so when you look through the grid of the cube you don't see raw wood!)

Ditto what SherryBinNH says about the base and painting it before attaching it to the existing ottoman.

I'd paint them in a high-gloss lacquer finish, with a fabric to pick up whatever color you paint them - something bright! They're very Jonathan Adler! You might look at his website for fabric and color inspiration www.jonathanadler.com. He uses lots of bright bold colors (fuschia! orange! chartreuse!) with more "conservative" colors, like navy and white and black.

Painting the laminate won't be a very hardy finish, but you could always have a glass and mirror supplier cut a piece of glass to cover the top, if you go that route. Or you could upholster them and use them as ottomans with a tray on top of one or both. If the laminate top is detachable, you might consider going to a Habitat for Humanity ReStore and seeing if there are any stone remnants you like that you could have cut for tops, if the bases are structurally sound enough to support the weight.

Paint the bottoms and tops. Then sew square cushions that tie at the corners so they are removable. You can use them as ottomans/stools, or side tables plus floor cushions. Add casters to the bottom and you have a couple of very versatile pieces of furniture.

I'd simply remove the laminate and place a large glass top over the pair of them to make a rectangular coffee table.

Can you remove the tops without damaging the construction? I like the glass top idea over them as end tables or one rectangular coffee table. The glass would need rounded edges to echo the shape of these.

I see them in bright red like this-
http://www.flickr.com/photos/44713708@N00/3948661552/in/set-72157622103111178/

Thank you all for the wonderful suggestions! In response to a few of you, I have taken them apart fairly easily to clean them. I am usually fond of restoring things in their natural state, however, the bases are pretty beat up that I'm sure a fresh coat of paint will fix. I love the ideas of a glossy vibrant finish or a crisp white. I am still torn on whether or not to upholster them. I do like the idea of painting the laminate so I could keep the hard surface. I know absolutely nothing about melamine paint. Does it leave a glossy finish? Also, would it withstand the banging of toys or drinking glasses without chipping? I will collect all these wonderful ideas and hopefully turn them into something beautiful and send AT a pic of the results if they would be willing to do a before and after!

Oh, so cool! Painting will absolutely work - any color you want. I found two laminate cubes, about 20" square, one rose and one aqua (the 80's are calling - they want their furniture back). Anyway, I primed them (no sanding needed) and painted both of them with white rustoleum paint, all 6 sides. I used them together as a coffee table for a while, and now they are on our beach balcony as tables. Impervious to everything, including salt air/water. So, I'd definitely paint them. Do the base in one color, and the top in another.

I love the idea of these with painted bases and upholstered tops - you could still use them as tables if you stick a tray on top but I would adore them with a glossy red base and a black-and-white patterned fabric on the top. And then they'd be no danger to the kid with some padding on top!

Yup, you're nuts. In a good way.