She’s three months old. I’d like to give it to her when she turns 18. She deserves it, after all, she’s the cutest niece in the world.
Does it matter what kind of account? Is there a specific type of interest-bearing account, or will an ordinary savings account do?
Also, I don’t see the need to tell her parents (or my SO, for reasons that will be explained shortly) just now, but in the event of my death, I would of course like it to go her. Will this entail a will? Then let’s add on a question - how much does a will cost, is there a cheap and easy way to go about it, etc.?
*The reason I don’t want to tell them is it’s his niece, his family, and they will all tell me not to, that I don’t have to, etc. You know, the old Asian saw (and possibly present in many other families also) about being overly polite about money. The bottom line is I want to, and I’ve always wanted to be the doting aunt. And now I get my chance.
(If ya’ll want pictures, I suppose I can provide. 
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How about a nice TDAmeritrade or HSBC high yield account. Both companies offer them and have lists of others. Good on you for doing this.
How about EE savings bonds? You pay half the face value, they continue to accrue interest until they are cashed in (or 30 years goes by, whichever is first).
They are liquid, like a savings account (you can cash them at any bank), don’t carry fees, are easy to move around with you, exempt from state and local taxes… and pretty to look at. 
http://www.savingsbonds.gov/indiv/products/prod_eebonds_glance.htm
Not very sexy (and certainly you can do better interest-wise), but they are a fail-safe no-hassle investment, and very giftable.
Oh, I forgot: they can be willed. I inherited a nice-sized one from my grandma.
You need a lawyer to draw up a will if this is important to you. Let’s say you die. How is anyone to guess what you wanted otherwise?
Side note: Ever since I started law school, I have an absolute horror of dying intestate (without a will).
If you just want to open a regular account for her, open a Totten trust account (named after an old NY case). It’s basically an account in your name, in trust for her. You can put money in and take it out all you want, but if you die, it goes straight to her, bypassing probate. When she’s older, she could also put money into it. When I was a kid, I had a trust passbook savings account that my dad set up for me. I put my extra money (after buying neat stuff with my allowance) in there. I got the whole thing when I was 18. It’s an extremely easy thing to do–those kinds of accounts have been around for more than 100 years in NY.