If we look into the terms of the CC BY-SA 4.0, we see the Section 3(a)(1)(A) requirement to “retain” the Licensor's “identification”, “including by pseudonym if designated”. So you should retain the user's chosen name in the form that you received it.
However, this identification is not immutable.
For example, Section 3(a)(3) also requires: “If requested by the Licensor, You must remove any of the information required by Section 3(a)(1)(A) to the extent reasonably practicable.”
Since you're dealing with electronic records, changing the attribution information is reasonably practicable.
For example, Stack Overflow supports this by “dissociating” posts from a username when the account is deleted, replacing the alias/name with the numerical ID.
I think a narrow reading of the CC BY-SA 4.0 supports that you could retain the old username on the content licensed to you, at least until the user requests the attribution to be removed entirely.
But since you will already need the ability to edit attribution information on old content, it would definitely be compliant (and the courteous thing to do) to allow users to change their displayed names retroactively.
I don't think that the CC BY-SA would in any way prevent you from updating old attribution information, especially when this update is requested by the user/rightsholder themselves.
A change of username can be reasonably understood as a request to update the attribution information on old content.
Note that the Creative Commons rules on attribution did not appear randomly.
These attribution requirements are essentially a contractual version of moral rights that are common in copyright laws (other than in US copyright law).
For example, German copyright law says:
The author has the right to be identified as the author of the work.
The author may determine whether the work is to bear a designation of authorship and which designation is to be used.
Note that this does not contain a limitation such as “at the time of publication”.
I think that France has similarly strong recognition for moral rights.
Users might change their legal name.
It's the internet.
Usernames might contain offensive or illegal content.
Systems should be built with the capability to moderate accordingly.
However, it would violate the CC BY-SA terms (and possibly moral rights) to change the username at your discretion, so that content from offending users would have to be removed entirely if they don't change their name “voluntarily”.
Difference in Version 2.
This answer assumes the CC BY-SA 4.0 license, since the question didn't specify a version.
However, the site in question seems to use CC BY-SA 2.0.
This version has different attribution requirements (see section 4(c)), and does not explicitly recognize the author's right to request attribution to be removed.
This does not affect whether the author has a right to remove or change their attribution per moral rights.