The DM cannot make your character care. That has to come from you.
A DM can do a lot to keep the PCs together, but there's only so far the DM can go before it straight-up becomes railroading. Since many groups (like mine) really don't like that, it falls to the players to give their characters at least some reason to keep the party together. I've been fortunate so far in that all my PCs have given me a reason for them to stick together, or at least given me something that I can use to keep them together sensibly. But if they hadn't done that, and any of them had decided their character had no reason not to leave, I can't really do much to stop that without stepping on player agency.
So what can you do?
Well, you have a few options.
1. Just have him leave.
One option, which you've already mentioned, is to have this character go ahead and leave, and roll up a new, more group-oriented character. Though this may not seem ideal, it does allow some interesting narrative opportunities down the road, as your pc now becomes an npc who is still active in the world. Crucially, it becomes an npc with personal ties to the party - it doesn't have to be anything incredibly deep to still be meaningful to that character. If the DM decides that he succeeds in saving his mother, this allows him to return. It also allows the DM to decide later that his quest eventually leads him to once again cross paths with the rest of the party.
A second option is to do as mxyzplk's answer suggests and change some aspect of your character to make him be more group-oriented. It may break your idea of the character, but frankly it sounds like you're already in that kind of situation. By forcing a loner type character with no bonds and no motive to stay in the group, you're already breaking the character. So, what do you do? Why, you meta-game!
"But, isn't meta-gaming bad?" I can hear people say, to which I reply, "SOMETIMES!" I do think that there is good and bad meta-gaming (and by "meta-gaming" I mean "using out-of-character knowledge to influence in-character actions and decisions"). Most people that I have talked to consider it bad form to use out-of-character knowledge to make things easier for the pc. Now, whether that kind of meta-gaming is good or bad is really up to the DM and the party as a whole, but that's a different conversation altogether and isn't at all the kind of meta-gaming I'm talking about.
What I'm talking about is meta-gaming used specifically to help make the game more fun for the players (including the DM). For example, it is (probably) more fun for both you and the party if everyone's character has a reason to travel together. That is out-of-character knowledge, but you can use that knowledge to make in-character decisions to make exactly that happen - whether by altering the character's background somehow or by intentionally making choices that make him become more and more attached to the other PCs. This kind of meta-gaming will keep you (the player) mindful of both your own and other players' (DM included - they're supposed to have fun too) enjoyment of the game, and is actually healthy and beneficial to the group as a whole.
And yes, as mxyzplk said earlier, this does also include recognizing and avoiding "My Guy" Syndrome.
As far as application of the principle, I might suggest making this character become attached to the party. Alternatively, you could have the character decide that, even if they're not moving as fast as he would like, that he obviously stands a better chance of rescuing his mother with the party at his side - perhaps he believes his enemies too dangerous to handle alone, and doesn't trust another adventuring group to be as capable? That way, you haven't even broken the character to make this decision!
3. Talk to your DM
Hobbamok already said it, but it never hurts to hear it again. Almost any problem in a table-top can be resolved by just talking with your DM and working out a solution with them.
And for that, I do have a few suggestions.
First, remember that it is not the DM's job to motivate your character. It is the DM's job to provide the setting and to determine the consequences of the party's actions. And to have fun, without stepping on someone else's fun. So you can ask the DM for help coming up with a stronger motivation for your character, but you cannot fairly ask the DM to come up with reasons and make your character care for you. That isn't why the DM is there, and they already have enough on their plate managing the world around the party.
So with that in mind, here is what I might do.
My first impulse would be to ask the DM to help me come up with a stronger reason to be in the party. I would come to the DM with suggestions, and let them decide on something that fits within their narrative. This might include discovering that her kidnappers are the same people the party is after, or it might involve those people killing her, turning your character's quest into a quest of personal revenge... aided by useful allies (a.k.a. The Party)! But... that second suggestion gets borderline Edge-Lord, and I don't know if that's the character you want to play.
You could also talk with the DM about the possibility of actually rescuing her before the end of the campaign, and then realizing that the Big Bad actually still poses a threat to her! That way, if your character wants to continue to protect her, he has to take down the Big Bad. This option would have the added bonus of letting the Party see him in a moment of tender vulnerability as he reunites with his mother, and those moments of vulnerability are typically what bring people closest together. Then, the DM has the choice of getting her safely out of the way while your character focuses on the long-term threat, or of having her decide to actively oppose the Big Bad, forcing your character to remain an active part of the group if he wishes to protect her.
But, don't expect the DM to come up with all of this on their own without you bringing it up. Remember, they're already creating the entire world your pc's interact with and are already playing every character that isn't a player character. That can get taxing, and it makes it difficult to come up with material to address one player character. So talk with your DM.
TLDR: You have to fix this
No matter what option you go with, you are the one who has to act on it. The DM can only do so much, and other players have no control over your character at all. Whatever action you take, whether it's to have the character leave, to change the character, or to work with the DM to give him better reasons to stay, it has to be initiated by you.