[Order of proposed names is strictly alphabetic. The supporting statements have been slightly edited for style]

1. Inductive Logic Programming (ILP)

The ILP community should work hard on changing the perception that many people have about ILP (synthesizing small logic programs - quicksort - from a few examples) to better match the broader research agenda that the community has been pursuing in the last 5 years or so AND to allow for furhter expansion. Example areas where we can reach out include learning probabilistic representations, relational data mining, hierarchical/relational reinforcement learning/planning, learning in natural language processing (LLL).
It may be that the name ILP does not adequately represent recent/contemporary research in the area. ILP may not have been the most appropriate name to begin with but it has stuck. People have heard about it, people know about it. It has taken  time, energy and enthusiasm of  many people to reach this stage and have, e.g.,a conference with Proceedings published by Springer.
A name change in and by itself would not change much in terms of attracting people to the meeting and the research area. In fact, it would probably have a harmful effect in terms of thecommunity losing its identity and visibility. I thus propose to keep the name ILP and actively work on reaching out to neighboring/related areas and communities according to the above plan.

2. Inductive Logic Programming and Multi-relational Data Mining (ILP-MRDM)

Multi-relational Data Mining is a good name for what many people do these days, and it is clearly much broader than ILP. It is better than "Relational Data Mining" because it excludes single relations (we don't want to subsume the rest of Machine Learning) and it hints a bit less strong at relational databases (which few approaches can actually interface with). By choosing a name of the form "ILP & X" we show confidence and realism about how the field develops, whereas dropping ILP completely would give the wrong signal that this is an emergency measure trying to rescue a dying field.

3. Inductive Logic Programming and Multi-Relational Knowledge Discovery (ILP-MRKD)

ILP should be present in the name for reasons of continuity and for people who understand it in the broader sense. The MRKD part of the name could act as an invitation to those who do not (yet?) understand this field as subsumed by ILP. Furthermore, KD could be more appealing than DM.

4. Induction of Logical and Probabilistic Relations (ILPr)

The lower-case "r" is an intentional pun on the Pr(x) function.
The reasons for supporting this name change are as follows.
a) It maintains contact with existing name recognition of "ILP" while making it clear that there is something new.
b) The change from "ILP" to "ILPr" should bring new people in from a significant and exciting new area within AI, ie the learning of probabilistic relations. ILPr could become the main conference for this area.
c) The "&" in various other suggestions is not a logical "and", but rather a political "and". This is undignified.

5. Inductive Logic Programming and Relational Data Mining (ILP-RDM)

The ILP community mainly consists of European and Japanese researchers. US researchers don't consider themselves to be  ILPers. A changed workshop name ILP&RDM would give a clear  indication of broadening the field to new areas, including US research on "link discovery" (part of the new DARPA project).  US researchers see themselves doing Relational data  mining or Relational learning or ... but not ILP.
Why adding RDM and not MRDM: (a) relational databases (RDB) are not called multi-relational databases (MRDB), but deal with multiple relations (b) RDM is similar to "Relational Learning": it can be interpreted also as "learning OF relations", not just "learning FROM relational data" (in this interpretation, MRDM would correspond  to multi-predicate learning) (c) RDM is more elegant and could catch better than MRDM (d) US people I talked to (Mooney, Page, Shavlik) liked the RDM proposal (e) Relational Data Mining book (Springer 2001, edited by Dzeroski and Lavrac) can be seen as an active step towards broadening the field.

6. Inductive Logic Programming and Relational Learning (ILP-RL)

Relational Learning is a broader concept than ILP: it accepts learning in any format that represents relations, including, but not limited to, logic, e.g. logic programs, graph representations, Sowa's conceptual graphs, probabilistic networks, etc. The term 'relational learning' has an established meaning outside Europe, and modifying the conference name would potentially extend it to a broader community. Last but not least, keeping ILP as part of the name would stress continuity and carry onthe existing connotation.

7. Induction of Logic Programms for Relational Learning and Mining of Structured Data (ILP)

On the one hand,  the best solution would be to keep our well etablished brand name (ILP) instead of creating a new unknown one. On the other hand, we have to break the unwanted (and fortunatly no longer really used) association with  synthesis of toy logic programs.
So the proposal is to keep the short acronym "ILP" unchanged and without any addition, but find a better fitting long title, e.g. "Induction of Logic Programms for relational learning and mining of structured data" or anything else (e.g. 2, 3, 5 or 6) that gives enough justification to keep ILP as shortname.

8. Learning from Structured Data (LSD?)

This title  describes the problem to be solved, not the technology that might be used to solve it. It also describes the problem well given that only a few words can be used - we are interested in learning problems for which the data is complex and  structured, and this requires new techniques. Generally, it's much better to name subfields by the major overall problem (e.g., computer vision,  machine learning, ...) rather than by a particular "religious" viewpoint (e.g., logic programming, inductive logic programming, ...) that implies some commitment to a particular technical approach that may  in time prove inadequate. Also lots of machine learning researchers study  structured data outside the confines of ILP. Finally, it should  be noted that the phrase `learning from data' is widely used; we are adding the extra constraint that the data be structured.

9. Relational Induction.

That covers it all, and there is no interpretation of relations without logic, nor is there any misinterpretation (as some suggest) that "relational" means single relations...that would be an interpretation taken by a very smallgroup, I think.