This is a tough one, as you can see by the varying replies below.
Does she go to a church? Is it possible maybe to go to her pastor or priest or rabbi and enlist her or his help with this? Many houses of worship have "benevolent funds" they use to give money to members or to perfect strangers who ask for help. I know that in the past, people in our church have gone to the pastor, and said just what you say here -- I want to give money anonymously to this person I know is in need. Then the pastor would take the money, contact the person, and say that a friend has donated money to the benevolent fund specifically earmarked for her and her kids, and wishes to remain anonymous.
That takes out the "Ick, it's stalkerish" factor that I do agree is present if you just mail her a card or stick cash into her wallet.
There is the chance she might refuse it and insist that the church keep it for church needs, especially if she is proud and feels that anonymous help is charity that she doesn't want to receive. If the pastor is a good one, he or she will be able to talk her into taking that money graciously and will not accept it for anything else at church.
That's just what occured to me. If your friend is not involved in a religious organization, or you don't feel secure about having a pastor involved like that, I would just come out and be open with her rather than anonymous. One important thing, if you do that: Be crystal clear that it's a gift and you never want her even to think of repaying it. Even if you say it's a gift she may be embarrassed by the large amount (a $1,500 gift is huge) and might try to repay you later anyway.
You know her personality best-- if she is the type who would feel she and you were both "saving face" by pretending that you "won this gift card" and are re-gifting it to her....Then that could work. I have friends who would see right through that instantly but also would say nothing and take the card if they were in need. I have other friends who would say, "Hey, why aren't you telling me the truth here? I know you didn't win this and I can't accept your charity" and it would turn into a struggle to give them money or a card despite the giver's good intentions. Think hard about your friend's personality and whether anonymous giving, or a fib about a found or won gift card, would be OK with her even if she sees through it, or whether it would just be best to be straightforward.
If she will just let you pay specific bills for her that might be easier on her own peace of mind that one big cash gift aimed at helping with Christmas She may feel (but not say she feels) hurt a bit that others know she can't provide much for her kids at the holidays, but might be more open to the cold business side of things like letting someone pay one month''s power bill as a gift to the whole family. Just something to consider as you think about her personality and the best way to give her something while maintaining your friendship. You are right --money, even as a gift with no strings, can alter things.
Also consider giving her maybe half or a third of the amount directly now and the rest at another time, in case she would find $1,500 to create a kind of pressure on her -- she might feel freaked at that high amount for a gift. Don't even casually mention "this is to help with the kids' holidays" etc. or she may feel she has to spend at least some of it that way when she may have greater needs you don't know about.
You are very generous and thoughtful and so is your husband for backing you up. I hope you update us on what you do and how it goes. Personally I'd go with the direct route, ask her whether she wants cash or a deposit into her checking account, and no commentary on what her needs might be.