Firstly I would like to note that there seems to be two reasonable versions of your question:

1. For which $X$ is $\#X(\mathbb{F}_p)$ is divisible by $p$ for any prime $p$?

2. For which $X$ is $\#X(\mathbb{F}_p)$ is divisible by $p$ for any sufficiently large prime $p$?

Both of these questions are quite interesting; yet the first one is more "arithmetic" and so harder to answer. Possibly I will treat it in a paper some day (thank you for asking it!).:) So, in this answer I will mostly treat the second question (which is also very hard).

Both of the questions are "motivic" since they depend on certain (complicated sort of) 'etale cohomology of $X$ only. Besides, they only depend on the class of the motif of $X$ (with compact support) with rational coefficients in the corresponding Grothendiek group of motives (that is also a ring). To get a sufficient condition for the first question one may consider the class of $X$ in a certain "complicate Grothendieck" group over the integers. For the second question it suffices to consider the motif with compact support of $X$ over $\mathbb{Q}$. This motif (and so, $X$ itself) has a well-defined class in the Grothendiek group of Chow motives (this is a seminal result of Gillet and Soule that answers a question of Serre; you may also have a look at my results on weight complexes and $K_0$(motives)). Thus $X$ also has a well-defined class in $K_0(Mot_{num})$. $X$ satisfies the conditions of Question 2 whenever this class is congruent to the one of a point modulo the Lefschetz motif (i.e., if $[X]-[pt]$ is the twist of an effective class).

At this stage one can ask two more natural questions:

3. It this $K_0$-condition a necessary one?

4. What geometric information does this condition contain?

I suspect that one can deduce the positive answer to Question 3 from certain "standard motivic" conjectures; yet this doesn't seem to be easy (and involves interaction between motives over fields of distinct characteristics; so one probably needs certain "relative motives" to achieve this).

Lastly, I would say that I do not expect any nice answers to Question 4. Certainly, the motivic conjectures predict that one can lift from $K_0(Mot_{num})$ to $K_0(Chow)$. Yet I see no way to rise from $K_0$ to motives themselves. Yet possibly I miss something here; then you may be interested in my results on Chow-weight homology: https://www.google.ru/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CB0QFjAA&url=http%3A%2F%2Farxiv.org%2Fabs%2F1411.6354&ei=dxWNVZbXO8XgyQOR8YCQCQ&usg=AFQjCNGZOey1IoXyWM5On9S2AUDnUTTmFA&sig2=nRumNYSvux2R2QblD1jcqQ&bvm=bv.96782255,d.bGQ&cad=rjt Anyway, the motivic assertions that one can obtain this way do not seems to imply anything like rational connectedness of varieties.

Firstly I would like to note that there seems to be two reasonable versions of your question:

1. For which $X$ is $\#X(\mathbb{F}_p)$ is divisible by $p$ for any prime $p$?

2. For which $X$ is $\#X(\mathbb{F}_p)$ is divisible by $p$ for any sufficiently large prime $p$?

Besides, in both of these questions one can demand $\#X(\mathbb{F}_q)$ to be divisible by $q$ for $q$ being any power of a prime. I will denote the corresponding versions of the questions by 1' and 2', respectively.

Both of these questions are quite interesting; yet Question 1 (and 1') is more "arithmetic" and so harder to answer. Possibly I will treat it in a paper some day (thank you for asking it!).:) So, in this answer I will mostly treat Questions 2 and 2' (that are also quite hard).

Both of the questions are "motivic" since (if we ignore a single prime $l$) they depend on $Rx_!\mathbb{Q}_{l,X}$ (considered as a mixed complex of $\mathbb{Q}_{l}$-etale sheaves over $Spec\, \mathbb{Z}[1/l]$; here $x:X\to Spec\, \mathbb{Z}$ is the structure morphism) only. Besides, they are of "Euler characteristic type". So, ("the prime-to-$l$-part of") Question 1' and Question 2' can be easily reformulated in terms of the class of $Rx_!\mathbb{Q}_{l,X}$ in the Grothendieck ring of mixed Galois representations of $Gal(\mathbb{Q})$. Are you interested in an answer of this type

Now I will proceed to motives (and avoid fixing $l$). The class mentioned only depends on the class of the motif of $X$ (with compact support) with rational coefficients in the corresponding Grothendiek group of motives (that is also a ring). To get a sufficient condition for the first question one may consider the class of $X$ in a certain "complicated motivic Grothendieck" group over the integers. For the second question it suffices to consider the motif with compact support of $X$ over $\mathbb{Q}$. This motif (and so, $X$ itself) has a well-defined class in the Grothendiek group of Chow motives (this is a seminal result of Gillet and Soule that answers a question of Serre; you may also have a look at my results on weight complexes and $K_0$(motives)). Thus $X$ also has a well-defined class in $K_0(Mot_{num})$. $X$ satisfies the conditions of Question 2' whenever this class is congruent to the one of a point modulo the Lefschetz motif (i.e., if $[X]-[pt]$ is the twist of an effective class).

At this stage one can ask two more natural questions:

3. It this $K_0$-condition a necessary one (also for Question 2)?

4. What geometric information does this condition contain?

I suspect that one can deduce the positive answer to Question 3 from certain "standard motivic" conjectures (certainly including the Tate one); yet this doesn't seem to be easy (though the Fontaine-Mazur condition could help here).

Lastly, I would say that I do not expect any nice answers to Question 4. Certainly, the motivic conjectures predict that one can lift from $K_0(Mot_{num})$ to $K_0(Chow)$. Yet I see no way to rise from $K_0$ to motives themselves. Yet possibly I miss something here; then you may be interested in my results on Chow-weight homology: https://www.google.ru/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CB0QFjAA&url=http%3A%2F%2Farxiv.org%2Fabs%2F1411.6354&ei=dxWNVZbXO8XgyQOR8YCQCQ&usg=AFQjCNGZOey1IoXyWM5On9S2AUDnUTTmFA&sig2=nRumNYSvux2R2QblD1jcqQ&bvm=bv.96782255,d.bGQ&cad=rjt Anyway, the motivic assertions that one can obtain this way do not seems to imply anything like rational connectedness of varieties.

Firstly I would like to note that there seems to be two reasonable versions of your question:

1. For which $X$ is $\#X(\mathbb{F}_p)$ is divisible by $p$ for any prime $p$?

2. For which $X$ is $\#X(\mathbb{F}_p)$ is divisible by $p$ for any sufficiently large prime $p$?

Both of these questions are quite interesting; yet the first one is more "arithmetic" and so harder to answer. Possibly I will treat it in a paper some day (thank you for asking it!).:) So, in this answer I will mostly treat the second question (which is also very hard).

Both of the questions are "motivic" since they depend on certain (complicated sort of) 'etale cohomology of $X$ only. Besides, they only depend on the class of the motif of $X$ (with compact support) with rational coefficients in the corresponding Grothendiek group of motives (that is also a ring). For the second question it suffices to consider the motif with compact support of $X$ over $\mathbb{Q}$. This motif (and so, $X$ itself) has a well-defined class in the Grothendiek group of Chow motives (this is a seminal result of Gillet and Soule that answers a question of Serre; you may also have a look at my results on weight complexes and $K_0$(motives)). Thus $X$ also has a well-defined class in $K_0(Mot_{num})$. $X$ satisfies the conditions of Question 2 whenever this class is congruent to the one of a point modulo the Lefschetz motif (i.e., if $[X]-[pt]$ is the twist of an effective class).

At this stage one can ask two more natural questions:

3. It this $K_0$-condition a necessary one?

4. What geometric information does this condition contain?

I suspect that one can deduce the positive answer to Question 3 from certain "standard motivic" conjectures; yet this doesn't seem to be easy (and involves interaction between motives over fields of distinct characteristics; so one probably needs certain "relative motives" to achieve this).

Lastly, I would say that I do not expect any nice answers to Question 4. Certainly, the motivic conjectures predict that one can lift from $K_0(Mot_{num})$ to $K_0(Chow)$. Yet I see no way to rise from $K_0$ to motives themselves. Yet possibly I miss something here; then you may be interested in my results on Chow-weight homology: https://www.google.ru/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CB0QFjAA&url=http%3A%2F%2Farxiv.org%2Fabs%2F1411.6354&ei=dxWNVZbXO8XgyQOR8YCQCQ&usg=AFQjCNGZOey1IoXyWM5On9S2AUDnUTTmFA&sig2=nRumNYSvux2R2QblD1jcqQ&bvm=bv.96782255,d.bGQ&cad=rjt Anyway, the motivic assertions that one can obtain this way do not seems to imply anything like rational connectedness of varieties.

Firstly I would like to note that there seems to be two reasonable versions of your question:

1. For which $X$ is $\#X(\mathbb{F}_p)$ is divisible by $p$ for any prime $p$?

2. For which $X$ is $\#X(\mathbb{F}_p)$ is divisible by $p$ for any sufficiently large prime $p$?

Both of these questions are quite interesting; yet the first one is more "arithmetic" and so harder to answer. Possibly I will treat it in a paper some day (thank you for asking it!).:) So, in this answer I will mostly treat the second question (which is also very hard).

Both of the questions are "motivic" since they depend on certain (complicated sort of) 'etale cohomology of $X$ only. Besides, they only depend on the class of the motif of $X$ (with compact support) with rational coefficients in the corresponding Grothendiek group of motives (that is also a ring). To get a sufficient condition for the first question one may consider the class of $X$ in a certain "complicate Grothendieck" group over the integers. For the second question it suffices to consider the motif with compact support of $X$ over $\mathbb{Q}$. This motif (and so, $X$ itself) has a well-defined class in the Grothendiek group of Chow motives (this is a seminal result of Gillet and Soule that answers a question of Serre; you may also have a look at my results on weight complexes and $K_0$(motives)). Thus $X$ also has a well-defined class in $K_0(Mot_{num})$. $X$ satisfies the conditions of Question 2 whenever this class is congruent to the one of a point modulo the Lefschetz motif (i.e., if $[X]-[pt]$ is the twist of an effective class).

At this stage one can ask two more natural questions:

3. It this $K_0$-condition a necessary one?

4. What geometric information does this condition contain?

I suspect that one can deduce the positive answer to Question 3 from certain "standard motivic" conjectures; yet this doesn't seem to be easy (and involves interaction between motives over fields of distinct characteristics; so one probably needs certain "relative motives" to achieve this).

Lastly, I would say that I do not expect any nice answers to Question 4. Certainly, the motivic conjectures predict that one can lift from $K_0(Mot_{num})$ to $K_0(Chow)$. Yet I see no way to rise from $K_0$ to motives themselves. Yet possibly I miss something here; then you may be interested in my results on Chow-weight homology: https://www.google.ru/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CB0QFjAA&url=http%3A%2F%2Farxiv.org%2Fabs%2F1411.6354&ei=dxWNVZbXO8XgyQOR8YCQCQ&usg=AFQjCNGZOey1IoXyWM5On9S2AUDnUTTmFA&sig2=nRumNYSvux2R2QblD1jcqQ&bvm=bv.96782255,d.bGQ&cad=rjt Anyway, the motivic assertions that one can obtain this way do not seems to imply anything like rational connectedness of varieties.