My two cents: bottom line she needs the cash, and fast. Don't worry about pride. Don't have her do your garden work if you're not hiring her because of her skills.

You can ask if there is anything you can do to help and don't be vague about wanting to give financial support. If she says she won't take your money, call it a loan. Make it very clear the the first priority is that this cat is healthy and the kid has a nice birthday. Money, pride, all that stuff comes not even second. Not even third.

The fact that she happened to mention the cost of the vet and that she won't be able to do anything for her sons birthday now is telling. No one with too much pride to accept money would divulge that.

Just be up front. Tell her how much you appreciate both her, and Stevie. Tell her since it's still so close to the holiday season, you hope she'll accept a holiday gift, on Stevie's behalf. Hand her the envelope, with the card and money. Let her do the typical "are you sure? you shouldn't..." and just insist, that you want her to have it. It's highly unlikely to go the wrong way.

EDIT: I personally wouldn't be the one to suggest a work trade, up front. That takes a "thoughtful gift" into the "charity" zone. If she offers however, then you can graciously accept.

Stevie is the perfect way to deflect but still be direct. "We love Stevie and would like to contribute to his recovery."

Wish we all had neighbors like you.

I think she would appreciate the $100 because you care about Stevie too. It's very touching and when you phrase it that way it's like you're helping the cat not her. Not patronizing at all

I think the idea of sending them a "get well soon" card for Stevie with some cash tucked in is a perfectly reasonable and subtle enough idea. Asking somebody to work for you to earn cash for their kids birthday because of unexpected expenses seems a bit on the nose to me but I'm not really familiar with the relationship I suppose.

Kindness isn’t patronizing. Just offer to help because you care for Stevie and your neighbor. No need to make it a debt (or worse, make her “work it off”), which would redefine your relationship as employer/employee. Just friends doing what friends do.

If this is real, you seem like a good person and I'm going to keep your example in mind.

I'm in a place where $200 isn't much, but if someone helped me help my kid that'd be priceless.

The way to help a person without being condescending is to add ZERO padding to your offer. Make it as direct as possible.

“Hey it makes us feel bad that you can’t afford your son’s birthday party now. Would it be okay with you if we provided the $200? It would really make us feel better”

No padding, no qualification, nothing. And frame it as what it is: a request for her to help you feel gratified by helping.

“Will you please let us pay? It would mean a lot to us” is how you need to present it to not be condescending.

Another post mentioned just giving cash anonymously and I think that's easily the best option. You would almost certainly have access to their mailbox if it's a suburban stand alone type? If not, an unmarked envelope under the door, with cash, would preserve plausible distance from making the neighbor feel like they have to decline out of etiquette.

Don't think about it more, they clearly need the help if they mentioned it, and if you can help without feeling the impact just do so without strings or direct attribution. They'll suspect it, and can if they approach you in genuine thanks if they want, then you're able to be gracious about accepting, or simply act surprised and happy that such a nice thing happened if not.

I've had people clearly embarrassed at the grocery checkout take a 50$ bill I claimed fell out of their pocket before several times. Preserves their dignity even if it's just a pretext for helping. Puts the ball in their court at least. "Hey man, I don't know what to say but it's not mine. Pay it forward for someone who needs help if it's not yours" is the worst that's ever gone for me before. Nobody likes being a charity case.

If she hasn't paid the bill yet. Call the vet and pay it.

Maybe use that fact that you found Stevie in your garden somehow? Something along the lines of "after we brought him to you we had a good look at our garden and we want to make some changes to make it easier to watch him when he plays out there" as an excuse for a short-term job for her perhaps?

Or maybe offer your own home or garden for a birthday event for the son with the excuse that the cat needs rest after such an incident and a party would be better placed elsewhere?

Great suggestions already offered, so I just want to thank you for such caring about Stevie and his owner.