The problem is that 'first french republic' cover a lot of actual regimes, between the fall of the monarchy and the establishment of the empire.
In order to keep a first french republic alive today, it needs to stay alive in its first years ( keeping it alive is just a matter of butterflies ).
There are two difficulties to that.
The first is internal revolt/coup. However, have basically anyone other than Napoleon succeed at a coup and the republic will be retained, if a restoration can be avoided. Coups by Hoche, Moreau or Massena are the usual suspects for these TL.
More interesting, IMO, are TL which avoid the need for these coup. OTL, the coup was a reaction against the excess of the directoire, which were, in turn, a reaction to the Terror. The most interesting PoD here, I think, are either the girondins win the power struggle ( which may require no invasion of France, or at least a delayed one ) or Robespierre stays in power ( he doesn't hesitate and sends the sections to arrest his foe instead of waiting and let them disperse ).
The second point is to avoid the kings retrun dans les fourgeons de l'ennemi. Avoiding this requires avoiding a successfull invasion of France. That is sucessfully uniting the whole of europe against France for a decade and fritting away all of the army for nothing. Just avoiding Napoleon will do this, I think.
Then, everything is up in the air.
You're coming up against the usual trouble with PoD during the french revolution ( and the reasons lazy author always get a french revolution exactly as OTL even with an earlier PoD ); it was nearly inevitable, but its OTL course was a very low probability course. Change anything and you change everything. And europe is no longer recognisable afterward, so you don't really have anything to build the rest of your TL upon.