Is there a good notion of holonomic $D$-modules on rigid analytic spaces?
Yes. Although it is only beginning to be developed.
You probably want to start with Berthelot: D-modules arithmétiques I : Opérateurs différentiels de niveau fini and Introduction à la théorie arithmétique des D-modules, and other papers that can be found at http://perso.univ-rennes1.fr/pierre.berthelot/ Section 5 of the second paper I mentioned is perhaps most relevant.
There is also a recent paper of Caro which I cannot find online called 'Holonomie sans structure de Frobenius et criteres d'Holonomie' which removes the necessity of the Frobenius action from Berhelot's work. I suppose he would send you a copy of upon request.
Finally, in a piece of shameless self-advertising, Konstantin Ardakov and I recently put a preprint on the arXiv http://arxiv.org/abs/1102.2606 part of which seeks to find a framework to further develop the theory.
Update: Caro's paper mentioned above now seems to be available here: http://aif.cedram.org/item?id=AIF_2011__61_4_1437_0 although you need a subscription to access it.
Further update (10th Feb 2015): Apologies for the further self-advertising but Konstantin Ardakov and I now have two further preprints on the topic of D-modules on rigid analytic spaces http://arxiv.org/abs/1501.02215 and http://arxiv.org/abs/1502.01273. There is no mention of holonomicity in either of these but there seems to be a natural definition of the notion in the framework outlined in these. Whether this definition behaves as one might hope is likely to be discussed in future work.
Further update (1st May 2019): Once again apologies for self-advertising but Konstantin Ardakov, Andreas Bode and I now have preprint https://arxiv.org/abs/1904.13280 with a tentative definition of holonomicity for a D-module on a rigid analytic space (D-module in the sense of my previous two papers with Ardakov mentioned in the last update).
For completeness I should also point to the paper La théorie du polynôme de Bernstein–Sato pour les algèbres de Tate et de Dwork–Monsky–Washnitzer by Mebkhout and Narváez-Macarro http://www.numdam.org/item/ASENS_1991_4_24_2_227_0/ that we cite in our latest preprint but I hadn't been previously aware of. This includes a more classical theory of D-modules on rigid spaces than ours --- the main distinction between the two theories is that our sheaves of rings are completions of theirs so that infinite order differential operators appear in our setting. Which theory might be preferred will depend on what problem is being considered.