1. It's difficult for me to tell which spaces are open for play and which aren't -- the border is hard to parse. (Is that this cell's border, or the adjacent cell?) This isn't really an issue during placement, but it is an issue for signalling that a cell has been "captured," which is hard for me to tell.
2. Hexagon grid is absolutely the right call. :)
3. The influence mechanic is very opaque: it's very hard for me to predict what will happen next, which makes me feel like I'm playing a little "intuitively," instead of planning out my moves with intention. Some way to see "this is the current state and this will be the next state" would help with that, but I think as long as influence is a continuous quantity it's going to be difficult to visualize and predict.
4. Making the board itself hexagonal (ala Hexxagon) will give you an odd number of cells, preventing ties. Might also be nice to smooth out cell-strength discontinuities around the top and bottom edges.
5. Balancing the first-player advantage by making the first move a "weak stone" was very confusing to me, because an adjacent move by player two will "capture" my initial stone with a single stone of theirs. I didn't understand this until I saw your comment about it here. I think the pie rule would work well in this game, but it's hard for me to reason about since edge moves are offensively weaker but defensively stronger... I'm not sure how well it would shake out, if you haven't tried it already.