A big problem with modern 'hint-based' genealogy is that algorithms for websites like Ancestry are heavily weighted to match people by name, and if the name in the computer index is not what you suspect, it is easy to miss them.
Rather than giving you a laundry list of websites you can try, let's start with some suggestions about different ways of searching the databases you already know about.
Review Your Prior research
Your question as written is clear and concise. For your own benefit, I suggest that you create two documents as part of your review of your prior research. First, make list of all source documents you have on hand. In a recent webinar, Daniel Earl suggested that we should sort our sources by the date the documents were created.
For example, in my own research, I have applications for delayed birth records for my great-uncles and great-aunts. Statewide birth certificates weren't available when they were born, and they applied for these now-standard records decades after they were born, most likely as part of the process to apply for a Social Security Card.
If you have documents where the creation date is not known, put them at the end of the list (as if the creation date were today's date). These "unknown date" documents might include a transcript or notes you've made about information passed down in the family, where you don't remember or didn't record the dates of the conversation with your family members.
Putting these documents in the order they were created rather than the order of the events referenced in them allows you to see at a glance how far removed the record is from the event it is describing.
Once you have your source list, create a timeline of events, with a cross-reference to the records in your source list which hold the information, so you know where the information came from (how you know what you 'know').
Understand the records that you are searching
Your next task is to understand the purpose for which the records were created, and the factors which might have influenced the information which was recorded on those records. I'll use the passenger lists as an example.
Passenger lists were created in the departure ports and have information which was given to steamship companies' ticket agents by the person who bought the tickets (the buyer was not necessarily the passenger). The information was recorded on intake forms which were copied to create the larger passenger manifest sheets that we are familiar with today. Like modern movie theatres, there were reduced-price tickets for children (see the GG Archives' article Booking Passage on a Steamship). Part of the reason so many of us may find children's passenger list entries with ages that don't quite match may be because their ages were nudged downward to get a better price, or because a relative bought the tickets and didn't know the children's ages as well.
Knowing nuances of record creation like these may help you when you are considering what information might appear in a database you're about to search. Be sure to follow any "Learn More" links or read "About the Database" information to learn more about the records. On FamilySearch, each record set has an article in the Research Wiki describing the scope of the record set and its limitations.
Understand the computer index and/or waypoints
The next technique that will prove helpful is to consider how the computer databases were created from the historical records. It may surprise some Ancestry users, to choose one example, to know that there are records in some Ancestry databases that can only be found via browsing or by a search for information that does not involve a name search. See Crista Cowan's 'Barefoot Genealogy video Some Genealogy Records Have No Name.
It is helpful to know what fields are available to be searched in any given database. Stephen P. Morse's One-Step Web Pages are designed with reminders for users searching the census and other records about what fields are actually in the record for particular years. (Tip: Reading the FAQs on Morse's site is a good way to get ideas for how to change up your searches.) Searching on the individual search page for a particular database or record set will usually give you more search options than using the generic 'global' search pages.
If you are browsing records on FamilySearch, read the Film Notes to understand how the records are arranged and how any bound volumes or packets of loose papers correspond to the rolls of microfilm. There will not be a neat one-to-one correspondence of book to film.
Understand the time and place
Consider the social history of the period to understand the records better, and to better evaluate the information you're going to look for. Know the localities in which your people lived (when you know it) and what records might have been created.
Record your search terms / Make a list of variants
Since you can't depend on finding your people by name, it's even more important to record what keywords and other search terms you used and what your results are. Review the records you've already collected and create a list of keywords and other information you might use for searching. Start with wide searches, paying attention to how many results you have, and note how the keywords reduce the number of records until you reach a set small enough to examine. Doing this will allow you to review what you have already tried. You won't be able to do complex searches in one setting, so expect to go back to this log again and again to say "Oh, I haven't tried this yet" as you think of new searches to try.
Search Techiniques and the Research Process
Genealogy databases on sites like Ancestry and elsewhere are generally set up so that we can look for people by name. For a problem like yours, a name search isn't helping. You've tried plugging in the 'known' date of birth for one of your research subjects into a site which is set up to find people that way and still had poor results? What next?
The assumption is that they changed their names again to distance
themselves from the family tragedy.
It's okay to say "what if" while you are searching, but be careful about making assumptions that will cause you to miss records. Write down any assumptions like this in your research plan or journal so you will have a record that this information or assumption came from you and not from any of the sources you've collected so far.
Try to break the habit of searching for people, which the big websites encourage us to do, and recognize that you are searching for records that might refer to your research subjects. In what records might you expect to find them? Write down your assumptions as "what ifs" -- questions, rather than statements of fact, and then think about how you might verify or find out more information.
Widen your search to include the FAN - family, associates, and neighbors. Have you searched for David and Esther in Ancestry's Army Transport Service Passenger lists where they might be listed as the nearest relatives to a soldier? Have you traced David Blistein and Esther Leah Cohen Blistein all the way to the end of life? Do you have obituaries for them, listing their survivors? Have you searched for probate records and other records created at the time of their death and after?
Do you know how Providence fared in the flu epidemic? Have you searched for reports of fatalities in the local newspapers? Ask the question "how many": How many people the same age as Hyman died in Provience during the epidemic? How many people the same age as Louis died in Boston?
If you think the boys changed their names, the obvious question to ask first is "Did they change their surname to Blistein?" to match their uncle's surname? Remember that except for the 1940 Census, where the census enumerator was asked to say who the informant was, we don't kow who talked to the census taker. It's possible that someone else told a clerk about "David and Esther Blistein and their boy Hyman" and he was recorded with their name.
Also, if you haven't already, try searching in every database by swapping the first name and surname fields. Every so often, you'll find indexes where the computer index is backwards, with the surname in the first name field, and the first name in the surname field. For databases where the indexes are made by Optical Character Recognition (OCR), entries can be skipped entirely, indexes can be a mashup of information about two different lines, surnames can persist in an index after the surname has changed on the printed page, and so on.
If you have the opportunity to search "the same" record set on multiple databases, use it, because the indexes may be independently generated. Search both Ancestry's Social Security Applications and Claims Index and the US National Archives' Access to Archival Databases (AAD)'s NUMIDENT database. Search for the parents in those databases as well, in case they were listed on the SS-5.
Search the US City Directories databases on both Ancestry and on MyHeritage. Ancestry's US City Directories databases are notoriously badly indexed. Use the feature to search for people at the same address on MyHeritage, and use keyword searching for the street name on Ancestry.
Search databases which have parallel information. Some men who are found in the WW1 Draft Registration cards also had to register during WW2. Have you searched the WW2 Draft Registrations for people with the same birth dates? Have you searched a year on either side in case the prospective draftee gave the same day or month of birth but a slightly different year?
Search newspapers in every locality where your subjects have relatives, and search widely in case reprints of obituaries or memorials show up in "10 years ago", "25 years ago" columns. It's not uncommon in small town papers for obituaries to be reprinted 50 years later.
Search thoroughly. Don't limit yourself to websites which are made for genealogists. You have lots of possible avenues to research: periodicals about marathon runners, Brown alumni, etc.