I like most of the suggestions you've gotten already. I was just at my cousin's son's 1st birthday party: talk about a carful of toys!!! So it is true they will have so many toys that they may not get used much. The savings bond is a great idea. However there are some great toys that stimulate them and can often grow with them.
If you want to get them a similiar toy so as not to play favorites, a push/pull/ride-on toy is great for this age (or tricycle). I also like the slide/outdoor toys idea. Here are few more ideas that I didn't see mentioned.
A sand box/ sand & water activity center and supplies for both. I'm partial to ones with covers- to keep the pests/pets out when not in use and the sand stays clean. My daughter absolutely LOVES playing with sand and dirt- so this way mommy knows it clean. http://www.toysrus.com/search/index.jsp?sr=1&f=Taxono...
Bounce & Spin Zebra: http://www.toysrus.com/product/index.jsp?productId=2298888
(They also have a pony which is interactive and can be hooked up to a TV. They had the pony set up at ToyRUs last time I was there, and my daughter walked right up to it and climbed on it with no problem and she's pretty tiny for her age. Most 1 year olds are her size or bigger.)
Imaginarium 5-way Tabletop Bead Maze: http://www.toysrus.com/product/index.jsp?productId=2674731
(I bought one for my daughter for Christmas- It is different has an abacus, xylophone, white magnetic board, bead maze and wire maze on top (like this one) and cost 2x as much. $60 was on sale for $40. I can't tell from pic, but her's looks like it might be bigger. It is a 12" cube and the wire beads on top can store inside by flipping upsidedown. She really loves it. She is now 18 months and goes to it often to entertain herself.)
There are few more great ones shown here- the activity tables/bead mazes: http://www.target.com/Bestsellers-12-18-Months-Baby-Toys/...
http://www.target.com/Battat-Parents-Kitty-Keyboard/dp/B0...
My sister got this for our little cousin. It is adorable! It says 2 years and up, but a 1 year old would really love it also and it doesn't have small parts.
Little People play sets and toys are great. http://www.toysrus.com/search/index.jsp?sr=1&kw=littl...
The Firetruck ball-pit looks really fun: http://www.toysrus.com/product/index.jsp?productId=3295287
Or maybe a ball popper. Some moms don't like the facts that the little balls can scatter, but then that's what happens with babies' toys- that's how they learn. I got my daughter's for her first birthday and she really loves it. http://www.target.com/Hasbro-Playskool-Busy-Ball-Popper/d... (I got mine online for under $20 and paid no shipping or taxes.)