I too am sorry for your loss.
I presume that your club would like to obtain your deceased member's call sign as a "vanity" (not sequentially issued) call sign for the club. The procedure is to first notify the FCC that your member has passed away, and request that the license be canceled. The ARRL has a web page that summarizes how to do so; you'll need some sort of proof, such as a copy of an obituary printed in the newspaper, or a copy of the death certificate. The request can be submitted online, by fax, or by mail. That web page says that the FCC typically take one to two weeks to cancel the license. You can discover whether the license has been canceled by checking the FCC ULS database online.
Once the license has been canceled, your club may submit a "vanity" call sign application request to the FCC. A club call signs is assigned to a trustee, that is a person who is responsible for the club's use of that call sign. A club can have any number of call signs, but for new applications, a person is only allowed to be a trustee for a single club call sign. In other words, the person listed on the application as the potential trustee must not already be the trustee of a club call sign.
The vanity call sign request can be submitted online. The FCC's web site has instructions. The proposed trustee will need to have a high-enough license class for the call sign; in other words, if the call sign is a 1x2, then the proposed trustee will need an Amateur-Extra-class license. The license application can be submitted online; the proposed trustee will need an FRN (FCC registration number).
The license application should specify the "in memoriam" provision, which indicates that the request is for the club to assume the call sign of the former member. If the application doesn't include that provision, then the request will be rejected. It would be a good idea for the club to get written permission from the surviving relative. The proof doesn't need to be submitted with the application, but it should be kept on file in case the application is disputed. (People have tried to abuse the vanity call sign system, skipping the usual two-year wait, by fraudulently obtaining call signs with the "in memoriam" provision before.)
To process vanity call sign applications, the FCC first wait 17 calendar days. Then, on the morning after the first federal work day thereafter, the FCC process the applications received on that day. (Source) Once the FCC's online database shows that the club call has been transferred, then you may use it.
Field Day this year is on the 23rd and 24th of June, 18 days from now. Unfortunately there won't be enough time for your club to get the call sign canceled and the vanity application processed before Field Day. You might as well get the ball rolling though, so that you can use the call sign for next year's Field Day!