Signing Time!!!! Definitely go with this please! (as an ASL signer with many family members, nieces and nephews who also sign as well as my own son)
I know you say it doesnt need to be the "official" sign language, but if you're going to learn and use something, you might as well use something that is understood by other people - if you make up your own signs then you have a babysitter or someone else or if she uses the sign later in life, no one will be able to understand her. It's hard to unlearn something once you're in the habit of using that sign.
Many of ASL's signs are simple and easy - those that aren't, baby hands will modify. For example "milk" is simply squeezing your fingers into a fist several times (think: getting milk out of a cow) Water is a little trickier, but my brother signed it with 2 fingers instead of 3 fingers (v instead of w) well into his tweens. Babies often sign it with just one finger, but that's "baby sign" just like "baby talk" when they say wa-wa instead of water.
Baby Signing Time (by Signing Time) is very very easy and fun - check the videos out from the library (around here, you have to put them on hold to get them. If they dont have it, ask them to order it.)
I hope that I (and others) dont overwhelm you with choices - signing is GREAT for baby(and anyone really!) My son didn't hit the terrible twos until he was 3 - when he wanted to do things that I wouldn't let him (like handle the knife, or go outside when it was naptime, etc.) Communication cuts out a lot of the frustration and tantrums that happens when a child has a specific thing mind and cannot express him/herself.
Another reason why I advocate Signing Time is because it is the only source I have ever seen people confident enough to use signs with me or my family outside of learning directly from us or a class. Signing Time accomplishes this, I think, by the way they present it (variety of signers, showing the word/object to be signed in a variety of ways, modeling that it's ok to speak without signing every single word, etc.) I only "discovered" signing time after people started signing with me and asking them where they learned it from. It was(is?) on TV as well as DVD's you can check out or buy.