Our DM did this in three campaigns using dreams and NPCs
Our first campaign figured a warlock, Pact of the Blade, Hexblade Aasimar whose Player Choice it was to not know who he was, where his parents were, nor where he came from. Yes, it was a bit of an edgelord move, but it worked well because the player (1) was consistent, (2) embraced the unknown and (3) enjoyed the twists and reveals along the way.
How can I give them clues to find it if the player and character
doesn't know anything about the person they are looking for?
His dreams. In our campaign, the dreams of each of the PCs informed story arcs that we could pursue or set aside. The key to this was good DM/Player interaction ahead of time, to give the DM time to weave the nuances of our character development into relevant plot arcs (and not just a bunch of side quests). That made buy-in to a given arc very easy for us, as players.
How many clues?
Each of us had a half dozen clues and pointers toward the unknown bits of our back story (or in my case a revenge arc). The puzzle pieces fell together as each character developed.
Also crucial to the success of this approach is for the DM and the players to have a good trust relationship - I feel that I must emphasize this. (It appears that you have that in your group).
Approach
He wove clues into dreams at first, then clues and tips appeared during some encounters (both combat and non-combat), and them some payoff sessions came along that were each focused on that character's arc. He did this with the other three of us and our backstories as well, even though we didn't play the "I have no idea who I am" card. We, as players, embraced this by supporting the other players/PCs during their mini arcs.
NPC interactions contribute heavily to this
In our hexblade's case, the Shadow realm powers made contact with him now and again; there was this mysterious cat who would crop up in dreams or during play only to disappear again. We ran into a cult in the mountains who ended up having clues as to who he had been previously.
The big reveal came around level 10 or 11, in a non-combat encounter, but right after a combat.
It need not take that long in your campaign: in campaign 2, this same "I don't know who my dad is" backstory was resolved quite a few levels before that. It involved a family signet ring as one of the major clues.
Make sure the player is on board with this
As you have that already covered, using NPCs (to include things like that cat our DM used) and dreams can provide clues along the way so that you can hold the reveal until the time is ripe. If the players guess ahead of time, all the better! (That happened in our second campaign).
The DM has previously posted here (@BenjaminTHall) on some D&D 4e and 5e topics, but I can't seem to find him here using a search. I have to give him a shout out here. He does what you are asking about extremely well; he is an excellent DM whose world building and style keeps us coming back for more. I ended up joining his games based on some conversations we had in GitP; all games have been on-line/VTT given the geographic dispersal of the players.