First, please accept our condolences for your loss.
Next, and I will have to get detailed here - whose HSA was it? HSAs belong only to an individual (like IRAs), so only one of you opened it (there is no such thing as a joint HSA).
Third, who had the HDHP insurance coverage? Just her? Or both of you? Did you (both) lose the HDHP coverage with her passing?
Fourth, if this was your wife's HSA, were you named as the beneficiary?
I don't see anything about a "death exception" for determining HSA eligibility. Where did you see that?
@dmertz is there anything you can to to assist?
If she was the HSA participant, I don't see how it would be permissible for her executor to make a contribution on her behalf to her HSA after her death. I would expect the reasoning to be largely the same as that that led to the IRS has stating that the executor is not permitted to make a regular IRA contribution on behalf of a decedent. Additionally, by operation of law the HSA ceases to be her HSA upon her death, either becoming her spouse if her spouse is the beneficiary or ceasing to be an HSA entirely if the beneficiary is not her spouse, so there would be no HSA that could receive any HSA contribution made on her behalf.
That's why I asked to whom the HSA belonged. It's not clear who opened the HSA and who had the HDHP coverage (and when). If the HSA was opened in the husband's name and he retained HDHP coverage, then he could certainly make a contribution, subject to the annual contribution limits.
We have seen many taxpayers who did not realize the "joint" HSA that they thought they had opened wasn't joint at all.
IRS Publication 969 mentions a death exemption. Does it apply?
The only thing I see is related to the "last month rule". If you started the HSA in January 2020, that does not apply to you.
You follow the usual rules. First, an HSA is owned by one person only. Second, if one spouse is covered by a family HDHP, then both spouses can contribute to an HSA as if both spouses had HDHPs even though the plan was only in 1 spouse's name.
So, if this is your spouse's HSA, no contributions are allowed after her death. Her maximum contribution (assuming that one or both of you was covered by a family HDHP and you had no other insurance that would disqualify you) would be $591, or $675 if she was age 55 or older. If you are now covered by a single HDHP since here death, your overall limit for 2020 would be $3845 minus whatever was contributed to her account, or $4845 minus whatever was contributed to her account if you are age 55 or older. (This is 11 months at single and 1 month at family.)
If the HSA is in your name, and assuming you had family coverage in January that switched to single coverage in February, then your limits would be $3845 or $4845, depending on your age.
"Second, if one spouse is covered by a family HDHP, then both spouses can contribute to an HSA as if both spouses had HDHPs even though the plan was only in 1 spouse's name."
That's not accurate. A person can contribute to an HSA only if that person is an HSA-eligible individual. One spouse being covered by a family HDHP plan does not necessarily make that individual's spouse an eligible individual. The family HDHP plan might cover one spouse and a child and that individual's spouse might not be covered by any HDHP plan or might be covered by Medicare. However, if both spouses are HSA-eligible with one spouse covered by a family HDHP plan and the other having self-only coverage, both spouses are considered to have family HDHP coverage. In other words, for anyone to contribute to an HSA, that individual must be covered by an HDHP plan and must not also be covered by some other, disqualifying coverage such as Medicare.
Without knowing who was covered by an HDHP plan (or will be covered if you change coverage) and whether the coverage was family coverage or self-only, it's impossible to be more specific about whether or not you are eligible to make HSA contributions to your own HSA.